Motion in One Dimension

Exercise 2(A) — Multiple Choice Type

Question 1

Physical quantities which are expressed only by their magnitude are :

1. Scalar quantities
2. Vector quantities
3. Tensor quantities
4. None of these.

Answer

Scalar quantities

Reason — Physical quantities are classified into two broad categories:

1. Scalar quantities - having only magnitude.
2. Vector quantities - having both magnitude and direction.

Question 2

Which of the following is a scalar quantity?

1. Force
2. Time
3. Acceleration
4. Displacement

Answer

Time

Reason — Time has only magnitude, whereas force, acceleration and displacement all have magnitude as well as direction.

Question 3

A vector quantity possesses:

1. Direction
2. Magnitude
3. Both (a) and (b)
4. None of these

Answer

Both (a) and (b)

Reason — Vector quantities have both magnitude and direction.

Question 4

A vector quantity is —

1. Work
2. Pressure
3. Distance
4. Velocity

Answer

Velocity

Reason — A vector quantity has both magnitude and direction. Velocity has both magnitude and direction, so it is a vector quantity.

Question 5

The motion of a train on a straight track is an example of ............... motion.

1. Two dimensional
2. Three dimensional
3. One dimensional
4. All of these

Answer

One dimensional

Reason — As the train is moving on a straight track and there is no lateral movement (sideways), hence it is a one dimensional motion.

Question 6

The S.I. unit of displacement is:

1. Centimetre
2. Millimetre
3. Decimetre
4. Metre

Answer

Metre

Reason — The shortest distance from the initial to the final position of the body, is the magnitude of displacement and its direction is from initial position to the final position. It is a vector quantity and its S.I. unit is metre.

Question 7

If a body starts its motion from point A to B and comes back to the same point after a certain time interval, the displacement is:

1. 10
2. 20
3. 0
4. 15

Answer

0

Reason — If a body starts its motion from point A to B and comes back to the same point after a certain time interval, the displacement is 0. For example: In a circular motion, the body comes back to the starting point after completing the circle, then displacement is zero but distance covered is not zero.

Question 8

Motion of a car in a crowded street is an example of:

1. Uniform speed
2. Uniform velocity
3. Variable acceleration
4. Uniform acceleration

Answer

Variable acceleration

Reason — As the change in velocity of the car in not same in same interval of time, the acceleration is said to be variable.

Question 9

The distance travelled by a body in 10 s when it travels with a uniform speed of 10 ms-1 is:

1. 100 m
2. 1 m
3. 20 m
4. 50 m

Answer

100 m

Reason — As the body is moving with uniform speed, equal distance is covered in equal intervals of time.

Distance covered in 1 s is 10 m, hence distance covered in 10 s is 10 x 10 = 100 m.

Question 10

For a particle in motion, which of the following quantity can be zero at any given instant?

1. Displacement
2. Distance
3. Speed
4. None of these

Answer

Displacement

Reason — If a body starts its motion from point A to B and comes back to the same after a certain time interval, the displacement is 0. Whereas, distance covered and speed will not be zero.

Question 11

The S.I. unit of velocity is —

1. km h^-1
2. m min^-1
3. km min^-1
4. m s^-1

Answer

m s^-1

Reason — Velocity is the rate of change of displacement (m) of a body with time (s) so its unit is m s^-1.

Question 12

The unit of retardation is —

1. m s^-1
2. m s^-2
3. m
4. m s²

Answer

m s^-2

Reason — Retardation is decline in velocity (ms^-1) of a body with time (s), so its unit is ms^-2. Retardation is negative value of acceleration and has same unit as acceleration.

Question 13

18 km h^-1 is equal to —

1. 10 m s^-1
2. 5 m s^-1
3. 18 m s^-1
4. 1.8 m s^-1

Answer

5 m s^-1

Reason — As,

$18 \text{ kmh}^{-1} = 18 \dfrac{\text{km}}{\text{h}} = \dfrac {18 \times 1000 m}{60 \times 60 \text{s}} \\[1em] \Rightarrow 18 \text{ kmh}^{-1} = \dfrac {18 \times 10 \text{m}}{6 \times 6 \text{s}} \\[1em] \Rightarrow 18 \text{\text{ kmh}}^{-1} = \dfrac {180 \text{ m}}{36 \text{ s}} \\[1em] \Rightarrow 18 \text{\text{ kmh}}^{-1} = 5 \text{ ms}^{-1}$

Hence, 18 km h^-1 is equal to 5 ms^-1.

Question 14

The value of g does not depend on the:

1. Height of the body
2. Mass of the body
3. Shape of the body
4. All of these

Answer

All of these

Reason — The value of g does not depend on height, mass or shape of the body. On the earth's surface, g is maximum at the poles and minium at the equator. The value of g decreases with altitude and also with depth from the earth's surface.

Question 15

Free fall of a body near earth's surface is an example of:

1. Uniform acceleration
2. Uniform velocity
3. Variable acceleration
4. None of these

Answer

Uniform acceleration

Reason — When a body is falling freely, the acceleration is said to be uniform as equal changes in velocity takes place in equal intervals of time.

Question 16

The value of g is maximum at:

1. Tropic of capricorn
2. Equator
3. Tropic of cancer
4. Poles

Answer

Poles

Reason — On the earth's surface, g is maximum at the poles and minium at the equator. The value of g decreases with altitude and also with depth from the earth's surface.

Question 17

A body when projected up with an initial velocity u goes to a maximum height h in time t and then comes back at the point of projection. The correct statement is —

1. The average velocity is 2h/t
2. The acceleration is zero
3. The final velocity on reaching the point of projection is 2u
4. The displacement is zero

Answer

The displacement is zero

Reason — The displacement is zero because the initial and the final position of the body is same.

Exercise 2(A) — Very Short Answer Type

Question 1

State whether the following quantity is a scalar or vector?

(a) Pressure

(b) Force

(c) Momentum

(d) Energy

(e) Weight

(f) Speed

Answer

(a) Pressure — Scalar quantity

(b) Force — Vector quantity

(c) Momentum — Vector quantity

(d) Energy — Scalar quantity

(e) Weight — Vector quantity

(f) Speed — Scalar quantity

Question 2

Write the parameters required to express a scalar quantity.

Answer

Two parameters are required to express a scalar quantity :

1. Unit in which the quantity is being expressed.
2. Numerical value of the measured quantity.

Question 3

Name the parameters required to express a vector quantity.

Answer

Three parameters are required to express a vector quantity :

1. Unit in which the quantity is being expressed.
2. Numerical value of the measured quantity.
3. Direction.

Question 4

What does the negative sign with a vector quantity indicates?

Answer

The negative sign with a vector quantity implies the reverse (or opposite) direction. e.g., the forces $\overrightarrow{F}$and $-\overrightarrow{F}$are in opposite directions.

Question 5

When is a body said to be at rest ?

Answer

A body is said to be at rest if it does not change its position with respect to its immediate surrounding.

Question 6

When is a body said to be in motion ?

Answer

A body is said to be in motion if it changes its position with respect to its immediate surroundings.

Question 7

What do you mean by motion in one direction ?

Answer

When a body moves along a straight line path, its motion is said to be one dimensional motion.

Question 8

Define displacement. State its unit.

Answer

The shortest distance from the initial to the final position of the body, is the magnitude of displacement and its direction is from initial position to the final position. It is a vector quantity.

Unit — The S.I. unit of displacement is metre (m) and C.G.S. unit is centimetre (cm)

Question 9

When is the magnitude of displacement equal to the distance?

Answer

The magnitude of displacement is equal to the distance, when the motion is in a fixed direction.

Question 10

Define velocity. State its unit.

Answer

The velocity of a body is the distance travelled per second by the body in a specified direction.

It is a vector quantity.

$\text{S.I. unit of velocity} = \dfrac{\text{metre}}{\text{second}} = \dfrac{\text{m}}{\text{s}} \\[0.5em] \text{C.G.S. unit of velocity} = \dfrac{\text{centimetre}}{\text{second}} = \dfrac{\text{cm}}{\text{s}} \\[0.5em]$.

Question 11

Define speed. What is its S.I. unit ?

Answer

Speed of a body is the rate of change of distance with time .

It is a scalar quantity. It is generally represented by u or v .

If a body travels a distance S in time t, then its speed v is —

$\text{Speed (v)} = \dfrac{\text{Distance (S)}}{\text{Time (t)}} = \dfrac{\text{S}}{\text{t}} \\[0.5em]$

$\text{S.I. unit of speed} = \dfrac{\text{metre}}{\text{second}} = \dfrac{\text{m}}{\text{s}} \\[0.5em] \text{C.G.S. unit of speed} = \dfrac{\text{centimetre}}{\text{second}} = \dfrac{\text{cm}}{\text{s}} \\[0.5em]$.

Question 12

Which quantity speed or velocity gives the direction of motion of a body.

Answer

Velocity gives direction of motion of body as it is a vector quantity. Speed is a scalar quantity hence, it does not say anything about direction of motion.

Question 13

When is the instantaneous speed same as the average speed?

Answer

In case of a body moving with uniform speed, the instantaneous speed and the average speed are equal (same as the uniform speed) because, in case of uniform speed, neither the speed nor the direction change.

Question 14

Define acceleration. State its S.I. unit.

Answer

Acceleration is the rate of change of velocity with time.

Numerically, acceleration is equal to the change in velocity in 1 s.

i.e.,

$\text{Acceleration} = \dfrac{\text{Change in velocity}}{\text{Time interval}} \\[0.5em] \text{S.I. unit of acceleration} = \dfrac{\text{meter}}{\text{second}^2} = \dfrac{\text{m}}{\text{s}^2}\\[0.5em] \text{C.G.S. unit of acceleration} = \dfrac{\text{centimeter}}{\text{second}^2} = \dfrac{\text{cm}}{\text{s}^2}\\[0.5em]$

Question 15

What is meant by the term retardation? Name its S.I. unit.

Answer

If the velocity of a body decreases with time, it is called retardation. As it is decrease in velocity per second, so, retardation is negative acceleration.

It's unit is same as that of acceleration.

$\text{S.I. unit of retardation} = \dfrac{\text{meter}}{\text{second}^2} = \dfrac{\text{m}}{\text{s}^2}\\[0.5em] \text{C.G.S. unit of retardation} = \dfrac{\text{centimeter}}{\text{second}^2} = \dfrac{\text{cm}}{\text{s}^2}\\[0.5em]$

Question 16

Which of the quantity, velocity or acceleration determines the direction of motion ?

Answer

Velocity determines the direction of motion as velocity is the distance travelled per second by a moving object in a particular direction.

Positive or negative sign of velocity indicates the direction of motion.

Acceleration is the rate of change of velocity with time. It does not say anything about direction of motion.

Positive or negative sign of acceleration tell us weather the velocity is increasing or decreasing.

Question 17

Define the term acceleration due to gravity. State its average value.

Answer

When a body falls freely under the gravity, the acceleration produced in the body due to earth's gravitational attraction is called acceleration due to gravity.

The average value of 'g' is 9.8 ms^-2 (or nearly 10 ms^-2).

Exercise 2(A) — Short Answer Type

Question 1

Differentiate between scalar and vector quantities, giving two examples of each.

Answer

Differences between scalar and vector quantities are as follows —

Question 2

Both scalar and vector quantities can be added, subtracted, multiplied and divided by simple arithmetic methods. Is this statement correct? Give a reason for your answer.

Answer

No, the statement is not correct. Scalar quantities can be added, subtracted, multiplied and divided by simple arithmetic methods however vector quantities follow different algebra for their addition, subtraction and multiplication as they have magnitude as well as direction.

Question 3(a)

Differentiate between distance and displacement.

Answer

The difference between distance and displacement is as follows:

Question 3(b)

Differentiate between speed and velocity.

Answer

The difference between speed and velocity is as follows:

Question 3(c)

Differentiate between uniform velocity and variable velocity.

Answer

The difference between uniform velocity and variable velocity is as follows:

Question 3(d)

Differentiate between average speed and average velocity.

Answer

The difference between average speed and average velocity is as follows —

Question 3(e)

Differentiate between acceleration and retardation.

Answer

The differences between acceleration and retardation are as follows —

Question 3(f)

Differentiate between uniform acceleration and variable acceleration.

Answer

The differences between uniform acceleration and variable acceleration are as follows:

Question 4

Can displacement be zero even if distance is not zero? Give one example to explain your answer.

Answer

Yes, the displacement can be zero, even if the distance is not zero.

If a body, after travelling, comes back to its starting point, the displacement is zero but the distance travelled is not zero.

Example — When a body is thrown vertically upwards from a point A on the ground, after some time it comes back to the same point A, then the displacement of the body is zero but the distance travelled by the body is not zero.

Question 5

Give an example of motion of a body moving with a constant speed, but with a variable velocity. Draw a diagram to represent such a motion.

Answer

The motion of a body in circular path, is an example of a body moving with a constant speed and variable velocity because the direction of motion of body changes continuously with time.

At any instant, the velocity is along the tangent to the circular path at that point.

The figure above shows the direction of velocity v at different points A, B, C and D of the circular path.

Question 6

Give an example of motion in which average speed is not zero, but average velocity is zero.

Answer

If a body starts its motion and comes back to the same point after a certain time, (e.g., in a circular motion,) the displacement is zero, so the average velocity is also zero, but the total distance travelled is not zero and therefore, the average speed is not zero.

Question 7

Give one example of each of the following:

(a) Uniform velocity

(b) Variable velocity

(c) Variable acceleration

(d) Uniform retardation

Answer

(a) Uniform velocity — The rain drops reach on earth’s surface falling with uniform velocity.

(b) Variable velocity — The motion of a freely falling body is with variable velocity because although the direction of motion of the body does not change, but the speed continuously increases.

(c) Variable acceleration — The motion of a vehicle on a hilly (or crowded) road.

(d) Uniform retardation — Motion of a vehicle, reaching a destination.

Question 8

The diagram below shows the pattern of the oil on the road, dripping at a constant rate from a moving car. What information do you get from it about the motion of car?

Answer

With the help of the given diagram, we observe that the red dots (dripping oil) are initially at regular intervals and later the gap between the dots decreases, which implies that initially the car was covering equal distances in equal intervals of time but later the distance is decreasing.

Thus, we can infer that initially the car is moving with a constant speed and then it slows down.

Question 9

What is the direction of velocity of an object moving in a circular path?

Answer

The direction of velocity of an object moving in a circular path is always tangent to the circle. This means that with the movement of body the velocity changes.

Question 10

'The value of g remains same at all places on the earth surface'. Is this statement true? Give reason for your answer.

Answer

No, the value of 'g' is not same at all places on the earth's surface. The value of g varies from place to place so an average value is considered. The average value of 'g' is 9.8 ms^-2 or nearly 10 ms^-2. The value of g decreases with altitude and also with depth from the earth's surface.

The value of 'g' is maximum at the poles and minimum at the equator on the earth surface.

Question 11

If a stone and a pencil are dropped simultaneously in vacuum from the top of a tower, which of the two will reach the ground first ? Give reason.

Answer

Both will reach the ground simultaneously as value of g does not depend on the mass of the body. Acceleration due to gravity is same ( = g ) on both and there is no effect of friction and buoyancy due to air.

Exercise 2(A) — Numericals

Question 1

The speed of a car is 72 km h^-1. Express it in m s^-1.

Answer

As,

$72 \text {km h}^{-1} = \dfrac{72 \text {km}}{1 \text {h}} = \dfrac {72 \times 1000 \text {m}}{(60 \times 60) \text {s}} \\[0.5em] \Rightarrow 72 \text {km h}^{-1} = \dfrac {72 \times 10 \text {m}}{(6 \times 6 )\text {s}} \\[0.5em] \Rightarrow 72 \text {\text {km h}}^{-1} = \dfrac {20 \times \text {m}}{\text {s}} \\[0.5em]$

Hence, 72km h^-1 is equal to 20 m s^-1.

Question 2

Express 15 m s^-1 in km h^-1 .

Answer

As,

$15 \text {m s}^{-1} = 15 \dfrac{\text {m}}{\text {s}} = \dfrac {15 \times 60 \times 60 }{1000} \\[0.5em] \Rightarrow 15 \text {m s}^{-1} = \dfrac {15 \times 6 \times 6 }{10} \\[0.5em] \Rightarrow 15 \text {m s}^{-1}= 54 \text {km h}^{-1} \\[0.5em]$

Hence, 15 m s^-1 is equal to 54 kmh^-1.

Question 3

Express each of the following in m s ^-1 —

(a) 1 km h^-1

(b) 18 km min^-1

Answer

(a) As,

$1 \text{km h}^{-1} = \dfrac{1 \text{km}}{1\text{h}} = \dfrac {1 \times 1000 \text{m}}{60 \times 60 \text{s}} \\[0.5em] \Rightarrow 1 \text{km h}^{-1} = \dfrac {1 \times 10 \text{m}}{6 \times 6 \text{s}} \\[0.5em] \Rightarrow 1 \text{km h}^{-1} = \dfrac {0.278 \times \text{m}}{\text{s}} \\[0.5em]$

Hence, 1 km h^-1 is equal to 0.278 ms^-1.

(b) As,

$18 \text{km min}^{-1} = 18 \dfrac{\text{km}}{\text{min}} = \dfrac {18 \times 1000 \text{m}}{60 \text{s}} \\[0.5em] \Rightarrow 18 \text{kmh}^{-1} = \dfrac {18 \times 10 \text{m}}{6 \text{s}} \\[0.5em] \Rightarrow 18 \text{kmh}^{-1}= \dfrac {300 \times \text{ m}}{\text{s}} \\[0.5em]$

Hence, 18 km h^-1 is equal to 300 m s^-1.

Question 4

Arrange the following speeds in increasing order — 10 m s^-1, 1 km min^-1, 18 km h^-1.

Answer

In order to arrange in increasing order of speeds, the units of the given speeds must be same.

Hence, we convert all the three in m s^-1.

(a) 10 m s^-1 = 10 m s^-1   .... [1]
(already in m s^-1)

(b) 1 km min^-1,

$\text{1 km min}^{-1} = \dfrac{\text{1 km}}{\text{1 min}} = \dfrac {1 \times 1000 \text{m}}{ 60 \text{s}} \\[0.5em] \Rightarrow \text{1 km min}^{-1} = \dfrac {1 \times 100 \text{m}}{6\text{s}} \\[0.5em] \Rightarrow \text{1 km min}^{-1} = \dfrac {16.66 \times \text{m}}{\text{s}} \\[0.5em]$

Hence,
1 km min^-1 = 16.66 m s^-1   .... [2]

(c) 18 km h^-1,

As,

$\text{18 km h}^{-1} = 18 \dfrac{km}{h} = \dfrac {18 \times 1000 m}{60 \times 60 s} \\[0.5em] \Rightarrow \text{18 km h}^{-1} = \dfrac {18 \times 10 m}{6 \times 6s} \\[0.5em] \Rightarrow \text{18 km h}^{-1} = \dfrac {5 \times m}{s} \\[0.5em]$

Hence,
18km h^-1 = 5 m s^-1   .... [3]

From 1, 2 and 3, we get the increasing order as follows —

18 km h^-1, 10 m s^-1, 1 km min^-1.

Question 5

A train takes 3 h to travel from Agra to Delhi with a uniform speed of 65 km h^-1. Find the distance between the two cities.

Answer

As we know,

Distance = Speed x Time

Given.

time = 3 h

speed = 65 km h^-1

Substituting the values in the formula above, we get,

Distance = 65 x 3 = 195 km

Hence, the distance between the two cities = 195 km .

Question 6

A car travels first 30 km with a uniform speed of 60 km h^-1 and then next 30 km with a uniform speed of 40 km h ^-1.

Calculate —

(i) the total time of journey,

(ii) the average speed of the car.

Answer

Given,

Distance travelled S₁ = 30 km

Speed v₁ = 60 km h^-1

Distance travelled S₂ = 30 km

Speed v₂ = 40 km h^-1

(i) As we know,

$\text {Time (t)} = \dfrac {\text {distance (S)}}{\text {speed (v)}}$

Substituting the values in the formula above, we get,

$\text{t}_1 = \dfrac{\text{S}_1}{\text{v}_1} \\[0.5em] \text{t}_1 = \dfrac{30 \text{km}}{60 \text{km h}^{-1}} \\[0.5em] \Rightarrow \text{t}_1 = \dfrac{1}{2} \text{h} \\[0.5em] \Rightarrow \text{t}_1 = 0.5 \text{h} \\[0.5em]$

Converting 0.5 h to min we get,

$\Rightarrow \text{t}_1 = 0.5 \times 60 \text{ min} \\[0.5em] \Rightarrow \text{t}_1 = 30 \text{ min} \\[0.5em]$

and

$\text{t}_2 = \dfrac{\text{S}_2}{\text{v}_2} \\[0.5em] \text{t}_2 = \dfrac{30 \text{km}}{40 \text{km h}^{-1}} \\[0.5em] \Rightarrow \text{t}_2 = \dfrac{3}{4} \text{h} \\[0.5em] \Rightarrow \text{t}_2 = 0.75 \text{h} \\[0.5em]$

Converting 0.75 h to min we get,

$\Rightarrow \text{t}_2 = 0.75 \times 60 \text{ min} \\[0.5em] \Rightarrow \text{t}_2 = 45 \text{ min} \\[0.5em]$

Total time t = t₁ + t₂
= 30 min + 45 min
= 75 min

Hence, Total time of journey = 75 min.

(ii) As we know,

Total distance S = S₁ + S₂
= 30 km + 30 km
= 60 km

Hence, Total distance travelled = 60 km.

As we know,

$\text{Average speed} = \dfrac{\text{total distance}}{\text{total time taken}} \\[0.5em] \text{Average speed} = \dfrac{60 \text { km}}{1.25 \text { h}} \\[0.5em] \Rightarrow \text{Average speed} = 48 \text { kmh} ^{-1}$

Hence, Average speed of the car = 48 kmh^-1

Question 7

A train takes 2 h to reach station B from station A, and then 3 h to return from station B to station A. The distance between the two stations is 200 km.

Find —

(i) The average speed

(ii) The average velocity of the train

Answer

Given,

Distance travelled S₁ = 200 km

time t₁ = 2 h

Distance travelled S₂ = 200 km

time t₂ = 3 h

(i) As we know,

$\text{Average speed} = \dfrac{\text{total distance}}{\text{total time taken}} \\[0.5em]$

Total distance travelled S = S₁ + S₂
= 200 km + 200 km
= 400 km

Total time taken t = t₁ + t₂
= 2 h + 3 h
= 5 h

Substituting the values of total distance travelled and total time taken in the formula of Average Speed above, we get,

$\text{Average speed} = \dfrac{400 \text { km}}{5 \text { h}} \\[0.5em] \Rightarrow \text{Average speed} = 80 \text { kmh} ^{-1} \\[0.5em]$

Hence, Average speed of the car = 80 km h^-1

(ii) Displacement = 0 (since final position is same as initial position)

Hence, Average velocity = 0 ( as displacement = 0)

Question 8

A car moving on a straight path covers a distance of 1 km due east in 100 s.

What is

(i) the speed and

(ii) the velocity of car ?

Answer

As we know,

$\text{Speed (v)} = \dfrac{\text{distance (S)}}{\text{time (t)}} \\[0.5em]$

Given,

Distance travelled S = 1 km = 1000 m,

time t = 100 s

Substituting the values in the formula above, we get,

$\text{Speed (v)} = \dfrac{1000 \text { m} }{ 100 \text{ s}} \\[0.5em] \Rightarrow \text{Speed (v)} = 10 \text { ms}^{-1} \\[0.5em]$

Hence, speed of the car is 10 m s^-1

(ii) The magnitude of velocity of the car is same as that of speed of car i.e., 10 m s^-1

However, when we talk about velocity, we mention direction also.

Hence, velocity of car is 10 m s^-1 due east.

Question 9

A body starts from rest and acquires a velocity 10 m s ^-1 in 2 s. Find its acceleration.

Answer

Given,

Initial velocity (u) = 0

Final velocity (v) = 10 m s ^-1

Time (t) = 2 s

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Substituting the values in the formula above, we get,

$\text {Acceleration (a)} = \dfrac {10 - 0}{2} \\[0.5em] \text {Acceleration (a)} = \dfrac {10\text{ ms}^{-1}}{2 \text{ s}} \\[0.5em] \text {Acceleration (a)} = 5 \text {ms}^{-2} \\[0.5em]$

Hence, acceleration of the body is 5 m s ^-2.

Question 10

A car starting from rest acquires a velocity of 180 m s^-1 in 0.05 h. Find the acceleration.

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Given,

Initial velocity (u) = 0

Final velocity (v) = 180 m s ^-1

Time (t) = 0.05 h

Converting 0.05 h to s we get,

$0.05 \text{h} = 0.05 \times 60 \times 60 \\[0.5em] 0.05 \text{h} = 0.05 \times 3600 \\[0.5em] 0.05 \text{h} = 180 \text {s} \\[0.5em]$

Substituting the values in the formula above, we get,

$\text {Acceleration (a)} = \dfrac {180 - 0}{180} \\[0.5em] \text {Acceleration (a)} = \dfrac {180 \text { ms}^{-1}}{180 \text { s}} \\[0.5em] \Rightarrow \text {Acceleration (a)} = 1 \text { ms}^{-2} \\[0.5em]$

Hence, acceleration of the body is 1 m s ^-2.

Question 11

A body is moving vertically upwards. Its velocity changes at a constant rate from 50 m s ^-1 to 20 m s ^-1 in 3 s . What is its acceleration ?

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Given,

Initial velocity (u) = 50 m s ^-1

Final velocity (v) = 20 m s ^-1

Time (t) = 3 s

Substituting the values in the formula above, we get,

$\text {Acceleration (a)} = \dfrac {20 - 50}{3} \\[0.5em] \text {Acceleration (a)} = - \dfrac {30\text { ms}^{-1}}{3\text { s}} \\[0.5em] \Rightarrow \text {Acceleration (a)} = -10 \text { ms}^{-2} \\[0.5em]$

Hence, acceleration of the body is -10 m s ^-2. Negative sign shows that the velocity decreases with time, so retardation is 10 m s ^-2.

Question 12

A toy car initially moving with a uniform velocity of 18 km h^-1 comes to a stop in 2 s. Find the retardation of the car in S.I. units.

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Given,

Initial velocity (u) = 18 km h^-1

To convert initial velocity u to m s^-1

$18 \text { km h}^{-1} = \dfrac{18 \text { km}}{1 \text{ h}} = \dfrac {18 \times 1000 \text{ m}}{60 \times 60 \text{ s}} \\[0.5em] \Rightarrow 18 \text { km h}^{-1} = \dfrac {18 \times 10 \text{ m}}{6 \times 6 \text{ s}} \\[0.5em] \Rightarrow 18 \text { km h}^{-1} = 5 \text{ m s}^{-1} \\[0.5em]$

Hence, 18 km h^-1 is equal to 5 m s^-1.

Final velocity (v) = 0

Time (t) = 2 s

Substituting the values in the formula above, we get,

$\text {Acceleration (a)} = \dfrac {0 - 5}{2} \\[0.5em] \text {Acceleration (a)} = -\dfrac {5\text { ms}^{-1}}{2\text { s}} \\[0.5em] \Rightarrow \text {Acceleration (a)} = - 2.5 \text { ms}^{-2} \\[0.5em]$

Hence, acceleration of the body is -2.5 m s ^-2. Negative sign shows that the velocity decreases with time, so retardation is 2.5 m s ^-2.

Question 13

A car accelerates at a rate of 5 m s^-2. Find the increase in its velocity in 2 s.

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Hence,

${\text {Velocity Increase}} = \text {Acceleration (a)} \times {\text{time(t)}}$

Given,

a = 5 m s^-2

t = 2 s

Substituting the values in the formula above, we get,

$\text {Velocity Increase} = 5 \text{ m s}^{-2} \times 2 \text{ s} \\[0.5em] \Rightarrow \text {Velocity Increase} = 10 \text{ m s}^{-1} \\[0.5em]$

Hence, increase in velocity = 10 m s ^-1.

Question 14

A car is moving with a velocity 20 m s ^-1 . The brakes are applied to retard it at a rate of 2 m s ^-2. What will be the velocity after 5 s of applying the brakes ?

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

and retardation is negative acceleration.

Hence,

$\text {final vel. (v)} - \text {initial vel. (u)} \\[0.5em] = \text {Retardation(-a)} \times {\text {time(t)}}$

Given,

a = - 2 m s^-2

t = 5 s

u = 20 m s ^-1

Substituting the values in the formula above, we get,

$\text {final vel. (v)} - 20 \text{m s}^{-1} = - 2 \text{m s}^{-2} \times 5 s \\[0.5em] \text {v} - 20 \text{m s}^{-1} = - 10 \text{m s}^{-1} \\[0.5em] \Rightarrow \text {v} = (20 - 10) \text{m s}^{-1} \\[0.5em] \Rightarrow \text {v} = 10 \text{m s}^{-1} \\[0.5em]$

Hence, final velocity = 10 m s ^-1.

Question 15

A bicycle initially moving with a velocity 5.0 m s ^-1 accelerates for 5 s at a rate of 2 m s^-2 . What will be its final velocity?

Answer

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Hence,

$\text {final vel. (v)} - \text {initial vel. (u)} \\[0.5em] = \text {Acceleration (a)} \times {\text {time (t)}}$

Given,

a = 2 m s^-2

t = 5 s

u = 5.0 m s ^-1

Substituting the values in the formula above, we get,

$\text {final vel. (v)} - 5 \text{m s}^{-1} = 2 \text{m s}^{-2} \times 5 s \\[0.5em] \text {v} - 5 \text{m s}^{-1} = 10 \text{m s}^{-1} \\[0.5em] \Rightarrow \text {v} = (10 + 5) \text{m s}^{-1} \\[0.5em] \Rightarrow \text {v} = 15 \text{m s}^{-1} \\[0.5em]$

Hence, final velocity = 15 m s ^-1.

Question 16

A car is moving in a straight line with speed 18 km h^-1. It is stopped in 5s by applying the brakes.

Find —

(i) the speed of car in m s ^-1,

(ii) the retardation and

(iii) the speed of car after 2 s of applying the brakes .

Answer

Given,

speed of car = 18 km h^-1

(i) To convert speed to m s^-1

$18 \text {km h}^{-1} = \dfrac{18 \text {km}}{1 \text{h}} = \dfrac {18 \times 1000 \text{m}}{60 \times 60 \text {s}} \\[0.5em] \Rightarrow 18 \text {km h}^{-1} = \dfrac {18 \times 10 \text {m}}{6 \times 6 \text {s}} \\[0.5em] \Rightarrow 18 \text {km h}^{-1} = 5 \text {m s}^{-1} \\[0.5em]$

Hence, 18 km h^-1 is equal to 5 m s^-1.

(ii) As the car is stopped, the final velocity = 0

initial velocity (u) = 18 km h^-1

As we know,

$\text {a} = \dfrac {\text {final vel. (v)} - \text {initial vel. (u)}}{\text {time (t)}}$

Given,

Initial velocity (u) = 5 m s^-1

Final velocity (v) = 0

Time (t) = 5 s

Substituting the values in the formula above, we get,

$\text {Acceleration (a)} = \dfrac {0 - 5}{5} \\[0.5em] \text {Acceleration (a)} = -\dfrac {5}{5} \\[0.5em] \Rightarrow \text {Acceleration (a)} = - 1 \text {ms}^{-2} \\[0.5em]$

Hence, acceleration of the car is -1 m s^-2. Negative sign shows that the velocity decreases with time, so retardation is 1 m s ^-2.

(iii) Given,

t = 2 s

a = -1 m s^-2

u = 5 m s^-1

Substituting the values in the formula for acceleration, we get,

$- 1 \text {ms}^{-2} = \dfrac {(v - 5 ) \text {ms}^{-1}}{2 \text {s}} \\[0.5em] - 2 \text {ms}^{-1} = (v - 5) {ms}^{-1} \\[0.5em] \Rightarrow v = (5 - 2) \text {ms}^{-1} \\[0.5em] \Rightarrow v = 3 \text {ms}^{-1} \\[0.5em]$

Hence, the speed of car after 2 s of applying the brakes = 3 m s^-1.

Give an example of motion of a body moving with a constant speed, but with a variable velocity. Draw a diagram to represent such a motion.

Answer

The motion of a body in circular path, is an example of a body moving with a constant speed and variable velocity because the direction of motion of body changes continuously with time.

At any instant, the velocity is along the tangent to the circular path at that point.

The figure above shows the direction of velocity v at different points A, B, C and D of the circular path.

Exercise 2(B) — Multiple Choice Type

Question 1

The slope of displacement-time graph gives:

1. Velocity
2. Acceleration
3. Speed
4. Displacement

Answer

Velocity

Reason — As, velocity is the ratio of displacement and time, therefore, the slope of displacement-time graph gives the velocity.

Question 2

From the given displacement-time graph, answer the following questions:

(i) The kind of motion depicted in this graph is :

1. uniform
2. non-uniform
3. retardation
4. all of the above

(ii) The velocity between point A and C is:

1. 20 ms^-1
2. 5 ms^-1
3. 10 ms^-1
4. 15 ms^-1

(iii) The displacement at t = 3s is:

1. 15 m/s²
2. 12.5 m/s²
3. 40 m
4. 30 m

Answer

(i) uniform
Reason — As the graph shows the linear relationship between displacement and time i.e., the car travels equal distance in equal intervals of time in a certain direction. Thus, it is moving with uniform motion.

(ii) 10 ms^-1
Reason — As, velocity is the ratio of displacement and time, therefore,

$\text{Velocity} = \text{Slope of straight line AC} \\[0.5em] = \dfrac{(60 - 30) \text{ m}}{(5 - 2) \text{ s}} \\[0.5em] = \dfrac{30 \text{ m}}{3 \text{ s}} \\[0.5em] = 10 \text{ m s}^{-1}$

(iii) 40 m
Reason — When we observe the graph, we notice that at t = 3 s, the displacement axis shows 40 m, hence, displacement at t = 3 s is 40 m

Question 3

Out of the following, the correct displacement-time graph for uniform motion is:

Answer

The correct displacement-time graph for uniform motion is:

Reason — When the displacement-time graph is a straight line inclined to the time axis, then the body covers equal distances in equal intervals of time and shows uniform motion.

Question 4

From the velocity-time graph, we can determine :

1. The displacement of the body in a certain time interval.
2. The acceleration of the body at any instance.
3. Both (a) and (b)
4. None of these

Answer

Both (a) and (b)

Reason — (i) Since, velocity x time = displacement, the area enclosed between the velocity-time sketch and X-axis (i.e., the time axis) gives the displacement of the body.

(ii) Since, acceleration is equal to the ratio of change in velocity and time taken, therefore the slope (or gradient) of the velocity-time sketch gives the acceleration.

Question 5

Velocity x time = :

1. Average velocity
2. Displacement
3. Acceleration
4. Speed

Answer

Displacement

Reason — Velocity x time = displacement.

Question 6

The velocity-time graph of a body in motion is a straight line inclined to the time axis. The correct statement is —

1. velocity is uniform
2. acceleration is uniform
3. both velocity and acceleration are uniform
4. neither velocity nor acceleration is uniform.

Answer

acceleration is uniform

Reason — When the velocity-time graph of a body in motion is a straight line inclined to the time axis then there are equal changes in velocity in equal intervals of time and hence, the movement is with uniform acceleration.

Question 7

For a uniformly retarded motion, the velocity-time graph is:

1. a curve
2. a straight line parallel to the time axis
3. a straight line perpendicular to the time axis
4. a straight line inclined to the time axis.

Answer

a straight line inclined to the time axis

Reason — If the motion is uniformly retarded (i.e., its velocity decreases by an equal amount in each second), the velocity-time graph will be a straight line inclined to the time axis with a negative slope.

Question 8

Study the velocity-time graph shown below and answer the questions that follow:

(i) The distance travelled in 5s:

1. 10 m
2. 50 m
3. 125 m
4. 250 m

(ii) The retardation of the body as calculated from the graph is:

1. 12 ms^-2
2. 15 ms^-2
3. 10 ms^-2
4. 20 ms^-2

Answer

(i) 125 m
Reason — As distance = velocity x time
So, (50 - 0) x (5 - 0) = 50 x 5 = 125 m
Hence, distance travelled in 5s = 125 m

(ii) 10 ms^-2
Reason —

$\text{Retardation} = \text{Slope of straight line} \\[0.5em] = \dfrac{(50 - 0)}{(5 - 0)} \\[0.5em] = \dfrac{50 \text{ m}}{5 \text{ s}} \\[0.5em] = 10 \text{ m s}^{-2}$

Hence, retardation = 10 ms^-2

Question 9

The graph representing the state of rest of an object is :

Answer

The graph representing the state of rest of an object is :

Reason — As distance is not changing in distance-time graph hence, the state of rest of an object is shown.

Question 10

Which of the following graphs shown below represents the uniform motion of an object?

Answer

The graph representing the uniform motion of an object is:

Reason — As in velocity-time graph, velocity of the body is not changing, therefore, the body is covering equal distance in equal intervals of time and is in uniform motion.

Question 11

The velocity-time graph given below shows:

1. Uniform acceleration
2. Average speed
3. Uniform velocity
4. All of these

Answer

Uniform velocity

Reason — As in velocity-time graph, velocity of the body is not changing, therefore, the body is covering equal distance in equal intervals of time and is in uniform motion.

Question 12

The slope of the velocity-tme graph gives :

1. Displacement
2. Speed
3. Distance
4. Acceleration

Answer

Acceleration

Reason — As, acceleration is the ratio of velocity and time, therefore, the slope gives the acceleration.

Question 13

In the velocity-time graph shown below, the ratio of the distance travelled by the object in the last 2 s and the distance travelled in 7 s is :

1. $\dfrac{1}{2}$

2. $\dfrac{1}{4}$

3. $\dfrac{1}{3}$

4. $\dfrac{2}{3}$

Answer

$\dfrac{1}{4}$

Reason — Distance covered in last 2 s = area of triangle = $\dfrac{1}{2}$ x 2 x 10 = 10 m

Distance covered in last 7 s = area of trapezium = $\dfrac{1}{2}$ x 10 x (2 + 6) = 40 m

Hence, ratio of of the distance travelled by the object in the last 2 s and the distance travelled in 7 s is = $\dfrac{10}{40}$ = $\dfrac{1}{4}$

Question 14

The velocity-time graph given below shows an object moving in a straight line. The displacement and the distance travelled by the object in 6 s will respectively be :

1. 8 m, 16 m
2. 16 m, 8 m
3. 16 m, 16 m
4. 8 m, 8 m

Answer

8 m, 16 m

Reason — Displacement = area of 1st rectangle - area of 2nd rectangle + area of third rectangle

= [4 x 2] - [2 x 2] + [2 x 2] = 8 m

Distance = area of 1st rectangle + area of 2nd rectangle + area of third rectangle

= [4 x 2] + [2 x 2] + [2 x 2] = 16 m

Exercise 2(B) — Very Short Answer Type

Question 1

For the motion with uniform velocity, how is the distance travelled related to the time?

Answer

For the motion with uniform velocity, distance is directly proportional to time.

Question 2

What does the slope of a displacement-time graph represent?

Answer

As, velocity is the ratio of displacement and time, therefore, the slope gives the velocity.

When slope is positive, it implies body is moving away from the starting (reference) point.

When slope is negative, it implies body is returning towards the starting (reference) point.

Question 3

When can the distance-time and displacement-time graph be the same?

Answer

For a motion in one direction in a straight line, since the direction of motion does not change, so displacement-time graph and the distance-time graph are same.

Question 4

What information do we get by a greater slope of displacement-time graph?

Answer

A greater slope of displacement-time graph shows higher velocity.

Question 5

What information is conveyed about motion by the negative slope of a displacement-time graph?

Answer

The negative slope of a displacement-time graph shows the motion towards the origin.

Question 6

What does the slope of velocity-time graph represent?

Answer

As the ratio of change in velocity and time taken is equal to the acceleration. Therefore, the slope of the velocity-time graph represents the acceleration.

Question 7

What information do we get by a higher slope of velocity-time graph?

Answer

On the velocity-time graph, larger the slope, higher is the acceleration or retardation.

Question 8

What information do we get about motion by the negative slope of a velocity-time graph?

Answer

If the velocity-time graph has a negative slope, the motion of body decreases at a constant rate (i.e., motion is uniformly retarded).

Exercise 2(B) — Short Answer Type

Question 1

What informations about the motion of a body are obtained from the displacement-time graph ?

Answer

In a displacement time graph, the time is on the X axis and the displacement of the body is on the Y axis. As, velocity is the ratio of displacement and time, therefore, the slope gives the velocity.

When slope is positive, it implies body is moving away from the starting (reference) point.

When slope is negative, it implies body is returning towards the starting (reference) point.

Question 2

Can displacement-time sketch be parallel to the displacement axis ? Give reason to your answer.

Answer

No, the displacement-time sketch can never be a straight line, parallel to the displacement axis because such a line would mean that the distance covered by the body in a certain direction increases without any increase in time (i.e., the velocity of the body is infinite) which is impossible.

Question 3

Draw a displacement-time graph for a boy going to school with a uniform velocity.

Answer

Below is the displacement-time graph for a boy going to school with a uniform velocity:

Question 4

State how the velocity-time graph can be used to find —

(i) the acceleration of a body,

(ii) the distance travelled by the body in a given time, and

(iii) the displacement of the body in a given time.

Answer

(i) Acceleration of a body — As acceleration is equal to the ratio of change in velocity and time taken, therefore, the slope ( or gradient ) of the velocity-time graph gives the acceleration.

(ii) Distance travelled by the body in a given time — The total distance travelled by the body is obtained by the arithmetic sum of the positive displacement and negative displacement (without sign).

(iii) Displacement of the body — As the product of velocity and time gives the displacement, therefore, the area enclosed between the velocity-time sketch and X - axis (i.e., the time axis) gives the displacement of the body.

The total displacement is obtained by adding positive displacement and negative displacement numerically with proper sign.

Question 5

What can you say about the nature of motion of a body if its displacement-time graph is —

(a) a straight line parallel to time axis?

(b) a straight line inclined to the time axis with an acute angle?

(c) a straight line inclined to the time axis with an obtuse angle?

(d) a curve

Answer

The following can be said about the nature of motion of a body, if its displacement-time graph is as follows —

(a) A straight line parallel to time axis — shows that the body is stationary (or no motion).

(b) A straight line inclined to the time axis with an acute angle — shows the linear relationship between the displacement and time (i.e., the body travels equal distance in equal intervals of time in a certain direction). Hence, we can say the motion is away from the starting point with uniform velocity.

(c) A straight line inclined to the time axis with an obtuse angle — shows that there is motion towards the starting point with a uniform velocity.

(d) A curve — shows that the body moves with varying speed in a fixed direction. Hence, we can say that there is motion with variable velocity.

Question 6

The figure shows displacement-time graph of two vehicles A and B moving along a straight road. Which vehicle is moving faster? Give reason.

Answer

We infer from the graph, that vehicle A is moving faster than vehicle B. It is so because, the slope of line A is more than that of line B.

Question 7

How is the velocity obtained from the displacement-time graph for a body moving with a varying speed in a fixed direction?

Answer

If a body moves with varying speed in a fixed direction i.e., with variable velocity, the displacement-time graph is not a straight line, but it is a curve. The velocity at any instant can then be obtained by finding the slope of the tangent drawn on the curve at that instant of time.

Question 8

Draw a velocity-time graph for a body moving with an initial velocity u and uniform acceleration a. Use this graph to find the distance travelled by the body in time t.

Answer

Below is the velocity-time graph for a body moving with an initial velocity u and uniform acceleration a:

Let v be the velocity at time t for this body having initial velocity u and uniform acceleration a as shown in the graph.

Distance travelled by the body in time t = Area under the velocity-time curve.

Hence,

Distance travelled = area of trapezium OABD

= $\dfrac{1}{2}$ x sum of parallel sides x height

= $\dfrac{1}{2}$ x (OA + DB) x OD

= $\dfrac{1}{2}$ x (u + v) x t

= $\dfrac{(\text{u} + \text{v})\text{t}}{2}$

Question 9

State the type of motion represented by the following sketches in (a) and (b).

Give example of each type of motion.

(a)

(b)

Answer

(a) The graph shows uniformly accelerated motion.

Example — Uniform acceleration is depicted by the motion of a body released downwards.

(b) The graph shows the motion with a variable retardation.

Example — Motion with a variable retardation is shown by a car approaching its destination.

Question 10

Figure shows the velocity-time graph for two cars A and B moving in same direction. Which car has the greater acceleration? Give reason to your answer.

Answer

We can infer from the graph, that the car B has greater acceleration than car A as the slope of straight line for car B is more than that of car A.

Question 11

Draw the velocity-time graph for a body moving with —

(a) uniform velocity,

(b) uniform acceleration.

Answer

(a) Below is the velocity-time graph for a body moving with uniform velocity:

(b) Below is the velocity-time graph for a body moving with uniform acceleration:

Question 12

The velocity-time graph for a uniformly retarded body is a straight line inclined to the time axis with an obtuse angle. How is retardation calculated from the velocity-time graph?

Answer

Retardation is calculated from the velocity-time graph, by finding the negative slope.

Question 13

The figure given below shows the displacement-time graph for four bodies A, B, C and D. In each case state what information do you get about the acceleration (zero, positive or negative).

Answer

The following information is obtained from the given figures —

• A → Zero acceleration since slope (i.e., velocity) is constant.
• B → Zero acceleration since slope (i.e., velocity) is constant.
• C → Negative acceleration (or retardation) since slope is decreasing with time.
• D → positive acceleration since slope is increasing with time.

Question 14

Draw a graph for acceleration against time for a uniformly accelerated motion. How can it be used to find the change in speed in a certain interval of time?

Answer

Below is the graph for acceleration against time for a uniformly accelerated motion:

For linear motion,

Acceleration x time = change in speed

Therefore, we can find the change in speed from the area enclosed between the acceleration-time sketch and the time axis.

Question 15

Draw a velocity-time graph for the free fall of a body under gravity, starting from rest.

Take g = 10 m s^-2.

Answer

Below is the velocity-time graph for the free fall of a body under gravity, starting from rest:

Question 16

How is the distance related with time for the motion under uniform acceleration such as the motion of a freely falling body ?

Answer

For a freely falling body (i.e., motion under uniform acceleration ), the displacement is directly proportional to the square of time.

                   S ∝ t²

Question 17

A body falls freely from a certain height. Show graphically the relation between the distance fallen and square of time. How will you determine g from this graph?

Answer

Below is the graph of distance fallen against square of time for a freely falling body:

The slope is half the acceleration due to gravity.

Thus, the value of acceleration due to gravity (g) can be obtained by doubling the slope of the S - t² graph for a freely falling body.

Exercise 2(B) — Long Answer Type

Question 1

Draw the velocity-time graph for a body moving with —

(a) uniform velocity,

(b) uniform acceleration.

and give one example of each and also calculate the displacement and acceleration.

Answer

(a) Below is the velocity-time graph for a body moving with uniform velocity:

Example: In the fig. given below, a straight line AB, represents the velocity-time graph of a body moving with a uniform velocity 4 ms^-1 for 5 s.

Displacement in 5 second = area of rectangle OABC = OC x OA = 5 x 4 = 20 m

Hence, displacement = 20m

The slope of the straight line AB is zero, therefore, its acceleration is zero.

(b) Below is the velocity-time graph for a body moving with uniform acceleration:

Example: The table below represents the velocity of a body at different instants, starting from rest.

Distance travelled in 8 s = S = area of triangle OPQ
= $\dfrac{1}{2}$ x base x height = $\dfrac{1}{2}$ x OQ x QP = $\dfrac{1}{2}$ x 8 x 80 = 320 m

Acceleration of body = slope of line OP
= $\dfrac{\text{PQ}}{\text{QO}}$ = $\dfrac{80 - 0}{8 - 0}$ = 10 ms^-2

Hence, distance covered is 320 m and acceleration = 10 ms^-2

Question 2

Draw the acceleration-time graph for a body moving with :

(a) Uniform velocity

(b) Free falling body

and give an example of each.

Answer

(a) Below is the acceleration-time graph for a body moving with uniform velocity:

In the above figure, if the body is moving with a uniform velocity, the acceleration is zero. Hence, acceleration-time graph in such a case is a straight line coinciding with time axis.

An example of a body moving with uniform velocity is a car travelling on a straight and flat road with a constant speed of 60 km/hour (or any other constant speed).

(b) Below is the acceleration-time graph for a free falling body:

In the figure given above, the straight line AF, represents the acceleration-time graph for a body falling freely with uniform acceleration of 10 ms^-2.

For example, a body falling freely under gravity moves with a uniform acceleration of 9.8 ms^-2 (or nearly 10 ms^-2).

Exercise 2(B) — Numericals

Question 1

Figure (a) shows the displacement-time graph for the motion of a body. Use it to calculate the velocity of body at t = 1 s, 2 s and 3 s, then draw the velocity-time graph for it in Figure (b).

Answer

We observe from the given displacement time graph above, that the slope is a straight line inclined with time axis, so the body is moving with uniform velocity. Hence, the velocity will be same at t = 1 s, 2 s and 3 s.

$\text{Velocity} = \text{Slope of straight line OP} \\[0.5em] = \dfrac{\text{AB}}{\text{OB}} \\[0.5em] = \dfrac{(2 - 0) \text{ m}}{(1 - 0) \text{ s}} \\[0.5em] = \dfrac{2 \text{ m}}{1 \text{ s}} \\[0.5em] = 2 \text{ m s}^{-1}$

Hence, velocity at t = 1 s, 2 s and 3 s is equal to 2 m s^-1

Velocity-time graph for the motion of a body is given below:

Question 2

Following table gives the displacement of a car at different instants of time.

(a) Draw the displacement-time sketch and find the average velocity of car.

(b) What will be the displacement of car at (i) 2.5 s and (ii) 4.5 s ?

Answer

(a) The displacement-time sketch for the car is shown below:

As we know,

$\text {Avg. velocity of car} = \dfrac {\text {Total distance}}{\text {Total time}} \\[1em] \text {Avg. velocity of car} = \dfrac {20 - 0 \text { m}}{4 - 0\text { s}} \\[1em] \text {Avg. velocity of car} = \dfrac {20 \text { m}}{4\text { s}} \\[1em] \Rightarrow \text {Avg. velocity of car} = 5\text { ms} ^ {-1}$

Hence, average velocity of the car is 5 ms^-1.

(b) When we observe the graph, we can find the displacement of the car at various points,

At t = 2.5 s, displacement = 12.5 m and

At t = 4.5 s, displacement = 22.5 m

Question 3

A body is moving in a straight line and its displacement at various instants of time is given in the following table —

Plot displacement-time graph and calculate —

(i) Total distance travelled in interval 1 s to 5 s,

(ii) Average velocity in time interval 1 s to 5 s.

Answer

The displacement-time graph is given below:

(i) When we observe the graph, we find —

Total distance travelled in interval 1 s to 5 s = 18 m - 6 m = 12 m

Hence, total distance travelled in interval 1 s to 5 s = 12 m

(ii) As we know,

$\text {Avg. velocity} = \dfrac {\text {Total distance}}{\text {Total time}} \\[1em] \text {Avg. velocity} = \dfrac {18 - 6 \text {m}}{5 - 1\text {s}} \\[1em] \text {Avg. velocity} = \dfrac {12 \text {m}}{4\text {s}} \\[1em] \Rightarrow \text {Avg. velocity} = 3\text {ms} ^ {-1}$

Hence, average velocity of the car in interval 1 s to 5 s is 3 ms^-1.

Question 4

Figure shows the displacement of a body at different times.

(a) Calculate the velocity of the body as it moves for time interval

(i) 0 to 5 s,

(ii) 5 s to 7 s and

(iii) 7 s to 9 s.

(b) Calculate the average velocity during the time interval 5 s to 9 s.

[ Hint — From 5 s to 9 s, displacement = 7 m - 3 m = 4 m ]

Answer

Below is the displacement-time graph of the body with the different points marked:

(a) As we know,

(i) At t = 0 to 5 s

Velocity = Slope of straight line OA

$\text {Velocity} = \dfrac {\text {AE}}{\text {OE}} = \dfrac {3 - 0 \text {m}}{5 - 0 \text {s}} \\[0.5em] \Rightarrow \text {Velocity} = \dfrac {3 \text {m}}{5 \text {s}} \\[0.5em] \Rightarrow \text {Velocity} = 0.6 {\text {ms}}^{-1} \\[0.5em]$

Hence, velocity at t = 0 to 5 s = 0.6 m s^-1

(ii) At t = 5 to 7 s

In this part we observe that there is no change in Y axis, (i.e. displacement is zero so the body is stationary).

Hence, velocity at t = 5 to 7 s = 0 m s^-1

(iii) At t = 7 s to 9 s

Velocity = Slope of straight line BC

$\text {Velocity} = \dfrac {\text {CD}}{\text {BD}} = \dfrac {7 - 3 \text {m}}{9 - 7 \text {s}} \\[0.5em] \Rightarrow \text {Velocity} = \dfrac {4 \text {m}}{2 \text {s}} \\[0.5em] \Rightarrow \text {Velocity} = 2 {\text {ms}}^{-1} \\[0.5em]$

Hence, velocity at t = 7 s to 9 s = 2 m s^-1

(b) As we know,

$\text {Avg. velocity} = \dfrac {\text {Total distance}}{\text {Total time}} \\[0.5em]$

Substituting the values from the graph we get,

$\text {Avg. velocity} = \dfrac {7 - 3 \text {m}}{9 - 5\text {s}} \\[0.5em] \text {Avg. velocity} = \dfrac {4 \text {m}}{4\text {s}} \\[0.5em] \Rightarrow \text {Avg. velocity} = 1\text {ms} ^ {-1} \\[0.5em]$

Hence, average velocity of the car is 1 ms^-1.

Question 5

From the displacement-time graph of a cyclist, given in figure, find —

(i) The average velocity in the first 4 s,

(ii) The displacement from the initial position at the end of 10 s

(iii) The time after which he reaches the starting point.

Answer

(i) As we know,

$\text {Avg. velocity} = \dfrac {\text {Total distance}}{\text {Total time}}$

Substituting the values from the graph, for the first 4 s, we get,

$\text {Avg. velocity} = \dfrac {10 \text {m}}{4 \text {s}} \\[0.5em] \Rightarrow \text {Avg. velocity} = 2.5\text {ms} ^ {-1} \\[0.5em]$

Hence, average velocity of the car is 2.5 ms^-1.

(ii) From the graph, we get,

Displacement = final position - initial position

Substituting the values from the graph, for the first 10 s, we get,

Displacement = -10 - 0 = -10 m

Hence, displacement from the initial position at the end of 10 s = - 10 m

(iii) The cyclist would reach the start point two times, one at the 7 s and the other at 13 s.

Question 6

Figure ahead represents the displacement-time sketch of motion of two cars A and B.

Find —

(i) The distance by which the car B was initially ahead of car A.

(ii) The velocities of car A and car B

(iii) The time in which car A catches car B

(iv) The distance from start when the car A will catch the car B

Answer

(i) When we observe the graph, we find that the car B was ahead of car A by 40 km.

(ii) As we know,

$\text{Velocity} = \dfrac{\text{displacement}}{\text{time}}$

Cars A and B have uniform velocities as displacement-time graph for both are straight lines.

At t = 4 h

Car A

$\text {Velocity} = \dfrac {160 - 0 \text {km}}{4 - 0 \text {h}} \\[0.5em] \Rightarrow \text {Velocity} = \dfrac {160 \text {km}}{4 \text {h}} \\[0.5em] \Rightarrow \text {Velocity} = 40 {\text {kmh}}^{-1} \\[0.5em]$

Hence, velocity of car A = 40 km h^-1

Car B

$\text {Velocity} = \dfrac {120 - 40 \text {km}}{4 - 0 \text {h}} \\[0.5em] \Rightarrow \text {Velocity} = \dfrac {80 \text {km}}{4 \text {h}} \\[0.5em] \Rightarrow \text {Velocity} = 20 {\text {kmh}}^{-1} \\[0.5em]$

Hence, velocity of car B = 20 km h^-1

When we observe the graph, we find that both cars A and B intersect at a point K.

Hence,

(iii) the car A catches car B at time (t) = 2 h.

(iv) the distance from start when the car A will catch the car B = 80 km.

Question 7

A body at rest is made to fall from the top of a tower. Its displacement at different instants is given in the following table —

Draw a displacement-time graph and state whether the motion is uniform or non-uniform?

Answer

Below is the displacement-time graph of the body made to fall from the top of a tower:

When we observe the graph, we find, that the displacement-time graph is a curve.

Hence, we can say that the motion is non-uniform.

Question 8

Figure shows the velocity-time graph for the motion of a body. Use it to find the displacement of the body at t = 1 s, 2 s, 3 s and 4 s, then draw the displacement-time graph for it on figure (b).

Answer

As we know, the displacement of body at any instant can be obtained by finding the area enclosed by the straight line with the time axis up to that instant.

Let displacements at t = 1 s, 2 s, 3 s, 4 s be S₁, S₂, S₃, S₄ respectively.

At t = 1 s,

S₁ = $\dfrac{1}{2}$ x 1 x 1 = 0.5 m

At t = 2 s,

S₂ = $\dfrac{1}{2}$ x 2 x 2 = 2 m

At t = 3 s,

S₃ = $\dfrac{1}{2}$ x 3 x 3 = 4.5 m

At t = 4 s,

S₄ = $\dfrac{1}{2}$ x 4 x 4 = 8 m

The table below gives the displacement of body at different instants.

The displacement time graph is shown below

Question 9

Figure below shows a velocity-time graph for a car starting from rest. The graph has three parts AB, BC and CD.

(i) State how is the distance travelled in any part determined from this graph.

(ii) Compare the distance travelled in part BC with the distance travelled in part AB.

(iii) Which part of graph shows motion with uniform (a) velocity (b) acceleration (c) retardation?

(iv) (a) Is the magnitude of acceleration higher or lower than that of retardation? Give a reason. (b) Compare the magnitude of acceleration and retardation.

Answer

(i) The distance travelled in any part of the graph can be obtained by finding the area enclosed by the graph in that part with the time axis.

(ii) Let E be the point on time axis (x axis) at time t and F be the point at time 2t.

When we observe the graph, we find,

Distance travelled in part BC = Area of rectangle EBCF

∴ Distance travelled in part BC = length x breadth
= (2t - t) x (v₀ - 0)
= t x v₀     .... [1]

Distance travelled in part AB = Area of triangle ABT

∴ Distance travelled in part AB = $\dfrac{1}{2}$ x base x height
= $\dfrac{1}{2}$ x (t - 0) x (v₀ - 0)

= $\dfrac{1}{2}$ x t x v₀     .... [2]

Comparing [1] and [2] we get,

Distance travelled in part BC : Distance travelled in part AB

$= \dfrac {t v_0}{\dfrac{tv_0}{2}} \\[0.5em] = \dfrac {2}{1}$

Hence,
Distance travelled in part BC : Distance travelled in part AB = 2 : 1

(iii) The different parts of the graph are mentioned below:

(a) Uniform velocity is shown in part BC of the graph, as the velocity is constant with time.

(b) Uniform acceleration is shown in part AB of the graph, as the velocity is increasing with time.

(c) Uniform retardation is shown in part CD of the graph, as the velocity is decreasing with time.

(iv) (a) The magnitude of acceleration is lower, as slope of line AB is less than that of line CD.

(b) Acceleration in part AB = slope of AB

$\text {Slope of line AB} = \dfrac{v_0 - 0}{t - 0} \\[0.5em] \Rightarrow\text {Slope of line AB} = \dfrac{v_0}{t} \\[0.5em]$

Retardation in part CD = slope of CD

$\text {Slope of line CD} = \dfrac{v_0 - 0}{2.5t - 2t} \\[0.5em] \Rightarrow\text {Slope of line AB} = \dfrac{v_0}{0.5t} \\[0.5em]$

Magnitude of acceleration : Magnitude of retardation = Slope of line AB : Slope of line CD

$= \dfrac{\dfrac{v_0}{t}}{\dfrac{v_0}{0.5t}} \\[1em] = \dfrac{0.5}{1} \\[1em] = \dfrac{5}{10} \\[1em] = \dfrac{1}{2}$

Hence, Magnitude of acceleration : Magnitude of retardation = 1 : 2

Question 10

The velocity-time graph of a moving body is given below in figure

Find —

(i) The acceleration in parts AB, BC and CD.

(ii) Displacement in each part AB, BC, CD, and

(iii) Total displacement.

Answer

Let E be the point at t = 4 and F be the point at t = 8 as labelled in the graph below:

(i) Acceleration in part AB = slope of AB

$\text {slope of AB} = \dfrac{(30 - 0) \text {ms}^{-1}}{(4-0) \text {s}} \\[0.5em] \text {slope of AB} = \dfrac{30 \text {ms}^{-1}}{4 \text {s}} \\[0.5em] \text {slope of AB} = 7.5 \text {ms}^{-2} \\[0.5em]$

Hence, Acceleration in part AB = 7.5 m s^-2

Acceleration in part BC = slope of BC

We observe from the graph that there is no change in velocity in part BC. Hence, Acceleration in part BC = 0 m s^-1

Acceleration in part CD = slope of CD

$\text {slope of CD} = \dfrac{(0 - 30) \text {ms}^{-1}}{(10-8) \text {s}} \\[0.5em] \text {slope of CD} = \dfrac{-30 \text {ms}^{-1}}{2 \text {s}} \\[0.5em] \text {slope of CD} = -15 \text {ms}^{-2} \\[0.5em]$

Hence, Acceleration in part CD = -15 m s^-1

(ii) Displacement in each part is as follows —

(a) Displacement of part AB = Area of triangle ABE

$= \dfrac {1}{2} \times \text {base} \times \text {height}$

Substituting the values in the formula above, we get,

$= \dfrac {1}{2} \times {(4 -0) \text {s}} \times (30 - 0)\text {m s}^{-1} \\[0.5em] = \dfrac {1}{2} \times 4 \text {s} \times 30 \text {m s}^{-1} \\[0.5em] = 30 \text {m} \\[0.5em]$

Hence,
Displacement of part AB = 60 m

(b) Displacement of part BC = Area of Square EBCF

$= \text {length} \times \text {breadth}$

Substituting the values in the formula above, we get,

$= {(8 -4) \text {s}} \times (30 - 0)\text {m s}^{-1} \\[0.5em] = 4 \times 30 \text {m} \\[0.5em] = 120 \text {m} \\[0.5em]$

Hence,
Displacement of part BC = 120 m

(c) Displacement of part CD = Area of triangle CDF

$= \dfrac {1}{2} \times \text {base} \times \text {height}$

Substituting the values in the formula above, we get,

$= \dfrac {1}{2} \times {(10 -8) \text {s}} \times (30 - 0)\text {m s}^{-1} \\[0.5em] = \dfrac {1}{2} \times 2 \times 30 \text {m} \\[0.5em] = 30 \text {m} \\[0.5em]$

Hence,
Displacement of part CD = 30 m

(iii) Total displacement = Displacement of part AB + Displacement of part BC + Displacement of part CD
= 60 + 30 + 120 = 210

Hence,
total displacement = 210 m

Question 11

A ball moves on a smooth floor in a straight line with a uniform velocity 10 m s^-1 for 6 s. At t = 6 s, the ball hits a wall and comes back along the same line to the starting point with same speed. Draw the velocity-time graph and use it to find the total distance travelled by the ball and its displacement.

Answer

The velocity-time graph for the motion of ball on a smooth floor in a straight line is shown below:

Total distance travelled = distance travelled in the first 6 s + distance travelled in next 6 s.

As we know,
Distance = velocity x time

Substituting the values in the formula above we get,

Distance when the ball moves towards the wall = 10 m s^-1 x 6 s = 60 m

Distance when the ball moves away from the wall = 10 m s^-1 x 6 s = 60 m

Total distance = 60 m + 60 m = 120 m

Displacement = distance travelled in the forward direction - distance travelled while coming back
= 60 m - 60 m = 0

∴ Displacement = 0 as the ball comes back to the initial place.

Question 12

Figure shows the velocity-time graph of a particle moving in a straight line.

(i) State the nature of motion of particle.

(ii) Find the displacement of particle at t = 6 s.

(iii) Does the particle change its direction of motion?

(iv) Compare the distance travelled by the particle from 0 to 4 s and from 4 s to 6 s.

(v) Find the acceleration from 0 to 4 s and retardation from 4 s to 6 s.

Answer

(i) As we observe the graph, we find that, the nature of motion of particle is that, the particles are uniformly accelerated from 0 to 4s and then uniformly retarded from 4s to 6s.

(ii) As we know,

displacement of particles can be obtained by finding the area enclosed by the graph in that part with the time axis up to that instance.

At t = 6 s,

Displacement = area of triangle

$= \dfrac {1}{2} \times \text {base} \times \text {height}$

Substituting the values in the formula above, we get,

$= \dfrac {1}{2} \times {(6 - 0) \text {s}} \times (2 - 0)\text {m s}^{-1} \\[0.5em] = \dfrac {1}{2} \times 6 \times 2 \text {m} \\[0.5em] = 6 \text {m} \\[0.5em]$

Hence,
Displacement of particle at t = 6 s is 6 m

(iii) No, the particle does not change its direction of motion.

(iv) At t = 0 to 4 s,

Distance covered = area of triangle

$= \dfrac {1}{2} \times \text {base} \times \text {height}$

Substituting the values in the formula above, we get,

$= \dfrac {1}{2} \times {(4 - 0) \text {s}} \times (2 - 0)\text {m s}^{-1} \\[0.5em] = \dfrac {1}{2} \times 4 \text {s} \times 2 \text {m s}^{-1} \\[0.5em] = 4 \text {m} \\[0.5em]$

Hence,
Distance covered between 0 to 4 s = 4 m

At t = 4 s to 6 s,

Distance covered = area of triangle

$= \dfrac {1}{2} \times \text {base} \times \text {height}$

Substituting the values in the formula above, we get,

$= \dfrac {1}{2} \times {(6 - 4) \text {s}} \times (2 - 0)\text {m s}^{-1} \\[0.5em] = \dfrac {1}{2} \times 2 \text {s} \times 2 \text {m s}^{-1} \\[0.5em] = 2 \text {m} \\[0.5em]$

Hence,
Distance covered between 4s to 6 s = 2 m

∴ Distance covered between 0 to 4 s : Distance covered between 4 s to 6 s = 4 : 2 = 2 : 1

(v) Acceleration in part 0 s to 4 s = slope of graph

$\text {slope} = \dfrac{(2 - 0) \text {ms}^{-1}}{(4-0) \text {s}} \\[0.5em] \text {slope} = \dfrac{2 \text {ms}^{-1}}{4 \text {s}} \\[0.5em] \text {slope} = 0.5 \text {ms}^{-2} \\[0.5em]$

Hence,
Acceleration in part 0 s to 4 s = 0.5 m s^-2

As we know,

Retardation in part 4 s to 6 s = slope of graph

$\text {slope} = \dfrac{(0 - 2) \text {ms}^{-1}}{(6 - 4) \text {s}} \\[0.5em] \text {slope} = \dfrac{- 2 \text {ms}^{-1}}{2\text {s}} \\[0.5em] \text {slope} = - 1 \text {ms}^{-2} \\[0.5em]$

Acceleration = -1 ms^-2 and as we know retardation is negative acceleration

Hence, Retardation in part 4 s to 6 s = 1 m s^-2

Exercise 2(C) — Multiple Choice Type

Question 1

The correct equation of motion is —

1. v = u + at
2. S = ut + $\dfrac{1}{2}$ at²
3. v² = u² + 2aS
4. All of these.

Answer

All of these

Reason — For motion of a body moving with a uniform acceleration, the following three equations give the relationship between initial velocity (u), final velocity (v), acceleration (a), time of journey (t) and distance travelled (S) :

1. v = u + at
2. S = ut + $\dfrac{1}{2}$ at²
3. v² = u² + 2aS

Question 2

When a body starts from rest, the equation of motion takes the form:

1. v = u
2. v = at
3. v = $\dfrac{1}{2}$at²
4. S = ut + $\dfrac{1}{2}$ at²

Answer

v = at

Reason — When a body starts from rest, initial velocity is zero (u = 0), then v = at.

Question 3

If a body is moving with a uniform retardation, then its acceleration will be:

1. Positive
2. Negative
3. Zero
4. Cannot say

Answer

Negative

Reason — When a body is moving with uniform retardation, then a will be negative.

Question 4

A car acquires a velocity of 54 ms^-1 in 20 s starting from rest, then its acceleration is:

1. 5.4 ms^-2
2. 2.7 ms^-2
3. 7.2 ms^-2
4. 2.0 ms^-2

Answer

2.7 ms^-2

Reason — Given,

u = 0
v = 54 ms^-1
t = 20 s
a = ?

Using, v = u + at or

a = $\dfrac{v - u}{t}$

a = $\dfrac{54 - 0}{20}$ = 2.7 ms^-2 .

Hence, acceleration = 2.7 ms^-2

Question 5

A particle starts to move in a straight line from a point with a velocity 10 ms^-1 and acceleration -2.0 ms^-2. Its position at t = 5 s will be :

1. 5 m
2. 10 m
3. 20 m
4. 25 m

Answer

25 m

Reason — Given,

u = 10 ms^-1
a = -2.0 ms^-2
t = 5 s

Using, S = ut + $\dfrac{1}{2}$at²

S = (10 x 5) + $\dfrac{1}{2}$ x (-2) x 5² = [50 - 25] = 25 m

Hence, its position at t = 5 s will be 25 m

Question 6

A body initially at rest, starts moving with a constant acceleration of 0.5 ms^-2 and travels a distance 25 m, then its final velocity is:

1. 5 ms^-1
2. 20 ms^-1
3. 15 ms^-1
4. -15 ms^-1

Answer

5 ms^-1

Reason — Given,

u = 0
a = 0.5 ms^-2
S = 25 m
v = ?

Using, v² = u² + 2aS

v² = 0 + [2 x 0.5 x 25] = 25

Hence, v = 5 ms^-1

Question 7

A body starts from rest with a uniform acceleration of 8 ms^-2, then the distance covered by the body in 2 s is:

1. 4 m
2. 16 m
3. 20 m
4. 32 m

Answer

16 m

Reason — Given,

u = 0
a = 8 ms^-2
t = 2 s
S = ?

Using, S = ut + $\dfrac{1}{2}$at²

S = (0 x 2) + $\dfrac{1}{2}$ x 8 x 2² = 16 m

Hence, the distance covered by the body is 16 m

Question 8

A car starting from rest accelerates uniformly to acquire a speed 20 km h^-1 in 30 min. The distance travelled by car in this time interval will be —

1. 600 km
2. 5 km
3. 6 km
4. 10 km

Answer

5 km

Reason —

As we know,

v = u + at

Given,

u = 0

v = 20 km h^-1

t = 30 min = $\dfrac{1}{2}$ h

Substituting the values in the formula above, we get,

$20 = 0 + (a \times \dfrac{1}{2}) \\[0.5em] \Rightarrow a = 40 \text{ km h}^{-2} \\[0.5em]$

Now,

S = ut + $\dfrac{1}{2}$ at²

Substituting the values in the formula above we get,

$\text{S} = (0 \times 0.5) + \Big[\dfrac{1}{2} \times 40 \times \Big(\dfrac{1}{2}\Big)^2\Big] \\[0.5em] \text{S} = 0 + \Big(\dfrac{1}{2} \times 40 \times \dfrac{1}{4}\Big) \\[0.5em] \Rightarrow \text{S} = 5 \text{ km} \\[0.5em]$

Hence, distance = 5 km

Exercise 2(C) — Very Short Answer Type

Question 1

Write three equations of uniformly accelerated motion relating the initial velocity (u), final velocity (v), time (t), acceleration (a) and displacement (S).

Answer

The equations of uniformly accelerated motion relating to the initial velocity (u), final velocity (v), time (t), acceleration (a) and displacement (S) are as follows —

(i) v = u + at

(ii) S = ut + $\dfrac{1}{2}$ at²

(iii) v² = u² + 2aS

Question 2

Write an expression for the distance S covered in time t by a body which is initially at rest and starts moving with a constant acceleration a.

Answer

According to equation of motion —

S = ut + $\dfrac{1}{2}$ at²

Where,

S = distance covered

u = initial velocity

a = acceleration

t = time

As, body is at rest, so u = 0

Hence, we get,

$S = (0\times t) + (\dfrac{1}{2} at^2) \\[0.5em] S = 0 + (\dfrac{1}{2} at^2) \\[0.5em] \Rightarrow S = \dfrac{1}{2} at^2 \\[0.5em]$

Hence, we get,

S = $\dfrac{1}{2}$ at²

Exercise 2(C) — Long Answer Type

Question 1

Derive the following equations for a uniformly accelerated motion —

(i) v = u + at

(ii) S = ut + $\dfrac{1}{2}$ at²

(iii) v² = u² + 2aS

where the symbols have their usual meanings.

Answer

(i) By definition,

$\text {Acceleration} = \dfrac {\text {change in velocity}}{\text {time taken}} \\[0.5em] \Rightarrow \text {Acceleration} = \dfrac {\text {final velocity - initial velocity}}{\text {time taken}} \\[0.5em]$

Hence,

$\text {a} = \dfrac{\text{v} - \text{u}}{\text {t}} \\[0.5em] \text {at} = \text{v} - \text{u} \\[0.5em] \Rightarrow \text {v} = \text{u} + \text{at} \\[0.5em]$

Hence, we get,

v = u + at     [Eq. 1]

(ii) By definition,

$\text {Distance travelled} = \text {Avg. Velocity} \times \text{time} \\[1em] \text {Distance travelled} = \Big(\dfrac{\text {Initial Vel.} + \text {Final Vel.}}{2}\Big) \times \text{time}$

or

$\text{S} = \Big(\dfrac {\text{u} + \text{v}}{2}\Big) \times \text{t}$

But, from Eq. 1, v = u + at

Therefore,

$\text{S} = \Big(\dfrac{\text {u + ( u +at)}}{2}\Big) \times \text {t} \\[0.5em] \text{S} = \Big(\dfrac{\text {2u + at}}{2}\Big) \times \text {t} \\[0.5em] \Rightarrow \text{S} = \text {ut} + \dfrac{1}{2} \text {at}^2 \\[0.5em]$

Hence, we get,
S = ut + $\bold{\dfrac{1}{2}}$ at²

(iii) By definition,

$\text {Distance travelled} = \text {Avg. Velocity} \times \text{time} \\[0.5em] \Rightarrow \text{S} = \Big(\dfrac {\text{u} + \text{v}}{2}\Big) \times \text{t}$

But, from Eq. 1, v = u + at

or t = $\dfrac{\text{v}-\text{u}}{\text{a}}$

Therefore,

$\text{S} = \Big(\dfrac {\text{v} + \text{u}}{2}\Big) \times \Big(\dfrac{\text{v}-\text{u}}{a}\Big) \\[1em] \Rightarrow \text{S} = \dfrac{\text{v}^2 - \text{u}^2}{2\text{a}}$

So, v² - u² = 2aS

Hence,

v² = u² + 2aS

Exercise 2(C) — Numericals

Question 1

A body starts from rest with a uniform acceleration of 2 m s^-2. Find the distance covered by the body in 2 s.

Answer

According to equation of motion —

S = ut + $\dfrac{1}{2}$ at²

Where,

distance covered = S

initial velocity u = 0

acceleration a = 2 m s^-2

time t = 2 s

Substituting the values in the formula for u = 0 , we get,

$S = (0 \times 2) + \dfrac{1}{2} \times 2 \times 2^2 \\[0.5em] S = (0) + (\dfrac{1}{2} \times 2 \times 4) \\[0.5em] S = (0) + (4) \\[0.5em] \Rightarrow S = 4 m \\[0.5em]$

Hence, distance covered = 4 m

Question 2

A body starts with an initial velocity of 10 m s^-1 and acceleration 5 m s^-2. Find the distance covered by it in 5 s .

Answer

According to equation of motion —

S = ut + $\dfrac{1}{2}$ at²

Where,

distance covered = S

initial velocity u = 10 m s^-1

acceleration a = 5 m s^-2

time t = 5 s

Substituting the values in the formula, we get,

$S = (10 \times 5) + (\dfrac{1}{2} \times 5 \times 5^2) \\[0.5em] S = (50) + (\dfrac{1}{2} \times 5 \times 25) \\[0.5em] S = (50) + ( 2.5 \times 25) \\[0.5em] S = 50 + 62.5 \\[0.5em] \Rightarrow S = 112.5 m \\[0.5em]$

Hence, distance covered = 112.5 m.

Question 3

A vehicle is accelerating on a straight road. Its velocity at any instant is 30 km h^-1, after 2 s, it is 33.6 km h^-1 and after further 2 s, it is 37.2 km h^-1. Find the acceleration of vehicle in m s^-2. Is the acceleration uniform ?

Answer

As we know,

a = $\dfrac{v - u}{t}$

After t = 2s,

u = 30 km h^-1

Converting km h^-1 to m s^-1

We get,

$30 \text{km h}^{-1} = \dfrac{30 \text{km}}{1\text{h}} = \dfrac {30 \times 1000 \text{m}}{60 \times 60 \text{s}} \\[0.5em] \Rightarrow 30 \text{km h}^{-1} = \dfrac {30 \times 10 \text{m}}{6 \times 6 \text{s}} \\[0.5em] \Rightarrow 30 \text{km h}^{-1} = \dfrac {8.33 \times \text{m}}{\text{s}} \\[0.5em]$

Hence, 30 km h^-1 is equal to 8.33 m s^-1.

Now,

v = 33.6 km h^-1

Converting km h^-1 to m s^-1

We get,

$33.6 \text{km h}^{-1} = \dfrac{33.6 \text{km}}{1\text{h}} = \dfrac {33.6 \times 1000 \text{m}}{60 \times 60 \text{s}} \\[0.5em] \Rightarrow 33.6 \text{km h}^{-1} = \dfrac {33.6 \times 10 \text{m}}{6 \times 6 \text{s}} \\[0.5em] \Rightarrow 33.6 \text{km h}^{-1} = \dfrac {9.33 \times \text{m}}{\text{s}} \\[0.5em]$

Hence, 33.6 km h^-1 is equal to 9.33 ms^-1.

Substituting the values in the formula above, we get,

$a = \dfrac{9.33 - 8.33}{2} \\[0.5em] a = \dfrac{1}{2} \\[0.5em] \Rightarrow a = 0.5 m s^{2} \\[0.5em]$

Hence, acceleration in the first 2 s = 0.5 m s ^-2

For the next 2 s,

u = 9.33 m s^-1

v = 37.2 km h^-1

Converting km h^-1 to m s^-1

We get,

$37.2 \text{km h}^{-1} = \dfrac{37.2\text{km}}{1\text{h}} = \dfrac {37.2 \times 1000 \text{m}}{60 \times 60 \text{s}} \\[0.5em] \Rightarrow 37.2 \text{km h}^{-1} = \dfrac {37.2 \times 10 \text{m}}{6 \times 6 \text{s}} \\[0.5em] \Rightarrow 37.2 \text{km h}^{-1} = \dfrac {10.33 \times \text{m}}{\text{s}} \\[0.5em]$

Hence, 37.2 km h^-1 is equal to 10.33 ms^-1.

Substituting the values in the formula above, we get,

$a = \dfrac{10.33 - 9.33 }{2} \\[0.5em] a = \dfrac{1}{2} \\[0.5em] \Rightarrow a = 0.5 m s^{2} \\[0.5em]$

Hence, acceleration is 0.5 m s ^-2

Yes, the acceleration is uniform as the acceleration in both instances is same.

Question 4

A body, initially at rest, starts moving with a constant acceleration 2 m s ^-2.

Calculate —

(i) the velocity acquired and

(ii) and the distance travelled in 5 s.

Answer

As we know, from the equation of motion,

v = u + at

Given,

a = 2 m s ^-2

u = 0

t = 5 s

Substituting the values in the formula, we get,

$v = 0 + (2 ms^{-2} \times 5 s) \\[0.5em] v = 10 ms^{-1} \\[0.5em]$

Hence, the velocity acquired = 10 m s^-1

(ii) According to equation of motion —

S = ut + $\dfrac{1}{2}$ at²

Substituting the values in the formula, we get,

$S = (0 \times 5) + (\dfrac{1}{2} \times 2 \times 5^2) \\[0.5em] S = 0 + (\dfrac{1}{2} \times 2 \times 5^2) \\[0.5em] \Rightarrow S = 0 + (\dfrac{1}{2} \times 2 \times 25) \\[0.5em] \Rightarrow S = 0 + 25$

Hence, distance travelled = 25 m

Question 5

A bullet initially moving with a velocity 20 m s^-1 strikes a target and comes to rest after penetrating a distance 10 cm in the target. Calculate the retardation caused by the target.

Answer

According to equation of motion —

v² = u² + 2aS

Given,

u = 20 m s^-1,

v = 0,

s = 10 cm = 0.1 m

Substituting the values in the formula above, we get,

$0 = (20)^2 + (2\times a \times 0.1) \\[0.5em] \Rightarrow 0 = 400 + 0.2 a \\[0.5em] \Rightarrow -400 = 0.2a \\[0.5em] \Rightarrow a = -\dfrac{400}{0.2} \\[0.5em] \Rightarrow a = -\dfrac{4000}{2} \\[0.5em] \Rightarrow a = - 2000 ms^{-2}\\[0.5em]$

Hence, Retardation is 2000 m s^-2

Question 6

A train moving with a velocity of 20 m s^-1 is brought to rest by applying brakes in 5 s. Calculate the retardation.

Answer

According to equation of motion —

v = u + at

Given,

u = 20 m s^-1,

v = 0,

t = 5 s

Substituting the values in the formula above, we get,

$0 = (20) + (a \times 5) \\[0.5em] 0 = 20 + 5a \\[0.5em] \Rightarrow -20 = 5a \\[0.5em] \Rightarrow a = -\dfrac{20}{5} \\[0.5em] \Rightarrow a = - 4 ms^{-2}\\[0.5em]$

Hence, Retardation is 4 m s^-2

Question 7

A train travels with a speed of 60 km h^-1 from station A to station B and then comes back with a speed 80 km h^-1 from station B to station A.

Find —

(i) the average speed, and

(ii) the average velocity of train.

Answer

As we know,

Avg Speed = $\dfrac{\text{Total Distance}}{\text{Total time}}$

Given,

From, A to B

speed_AB = 60 km h^-1

distance travelled = d_AB = d

time taken = $\dfrac{\text{d}}{60}$

From, B to A

speed_BA = 80 km h^-1

distance travelled = d_BA = d

time taken = $\dfrac{\text{d}}{80}$

Total distance travelled = d + d = 2d    [Eqn. 1]

Total time = $\dfrac{\text{d}}{60}$ + $\dfrac{\text{d}}{80}$    [Eqn. 2]

Substituting the values from the equations 1 and 2 in the formula we get,

$\text {Avg Speed} = \dfrac{\text {2d}}{\dfrac{\text {d}}{60} + {\dfrac{\text {d}}{80}}} \\[0.5em] \text {Avg Speed} = \dfrac{\text {2d}}{\dfrac {\text {4d + 3d}}{240}} \\[0.5em] \text {Avg Speed} = \dfrac{\text {2d}}{\dfrac {\text {7d}}{240}} \\[0.5em] \text {Avg Speed} = \dfrac{\text {2}}{\dfrac {\text {7}}{240}} \\[0.5em] \text {Avg Speed} = \dfrac {2 \times 240}{7} \\[0.5em] \text {Avg Speed} = \dfrac {480}{7} \\[0.5em] \Rightarrow \text {Avg Speed} = 68.57 \text{ km h}^{-1} \\[0.5em]$

Hence, Avg Speed = 68.57 km h^-1

(ii) We know,

Average velocity = $\dfrac{\text {displacement}}{\text {total time}}$

As the train starts from station A and comes back to same station, hence, displacement is zero.

Therefore, the average velocity is also zero.

Question 8

A train is moving with a velocity of 90 km h^-1. It is brought to stop by applying the brakes which produce a retardation of 0.5 m s^-2.

Find —

(i) the velocity after 10 s, and

(ii) the time taken by the train to come to rest.

Answer

(i) As we know, according to the equation of motion,

v = u + at

Given,

u = 90 km h^-1

velocity in m s^-1

$90 \text{km h}^{-1} = \dfrac{90 \text{km}}{1\text{h}} = \dfrac {90 \times 1000 \text{m}}{60 \times 60 \text{s}} \\[0.5em] \Rightarrow 90 \text{km h}^{-1} = \dfrac {90 \times 10 \text{m}}{6 \times 6 \text{s}} \\[0.5em] \Rightarrow90 \text{km h}^{-1} = \dfrac {15 \times 10 \text{m}}{1 \times 6 \text{s}} \\[0.5em] \Rightarrow 90 \text{km h}^{-1} = \dfrac {25 \times \text{m}}{\text{s}} \\[0.5em]$

Hence, 90 km h^-1 is equal to 25 m s^-1.

v = 0 (as the train stops on application of brakes)

t = 10 s

retardation = -a = -0.5 m s^-2

Substituting the values in the formula above we get,

$\text {v} = 25 + [(-0.5) \times 10] \\[0.5em] \text {v} = 25 + [- 5] \\[0.5em] \Rightarrow \text {v} = 20 \text {m s}^{-1} \\[0.5em]$

Hence, the velocity after 10 s = 20 m s^-1.

(ii) As we know, according to the equation of motion,

v = u + at

Substituting the values in the formula above, to get the time taken by the train to come to rest.

$\text {0} = 25 + (-0.5 \times \text{t}) \\[0.5em] \text {0} = 25 - 0.5 \text{t} \\[0.5em] 0.5 \text{t} = 25 \\[0.5em] \Rightarrow \text {t} = \dfrac {25}{0.5} \\[0.5em] \Rightarrow \text {t} = \dfrac {250}{5} \\[0.5em] \Rightarrow \text {t} = 50 \text {s} \\[0.5em]$

Hence, the time taken by the train to come to rest = 50 s.

Question 9

A car travels a distance 100 m with a constant acceleration and average velocity of 20 m s^-1. The final velocity acquired by the car is 25 m s^-1.

Find:

(i) the initial velocity and

(ii) acceleration of car.

Answer

(i) Given,

v = 25 m s^-1

As we know,

average velocity = $\dfrac{u+v}{2}$ = 20 m s^-1