Speed, Distance & Time - Exercise 12(B)

Which of the following is correct?

1. Distance = Speed x Time

2. Time = Distance x Speed

3. Speed = Distance x Time

4. Time = $\dfrac{\text {Speed}}{\text {Distance}}$

Answer

Distance = Speed x Time is correct.

All other options are incorrect.

Hence, option 1 is the correct option.

To convert a speed of km/hr into m/sec, we multiply it by

1. $\dfrac{8}{15}$

2. $\dfrac{18}{5}$

3. $\dfrac{5}{18}$

4. $\dfrac{15}{8}$

Answer

To convert a speed of km/hr into m/sec, we multiply it by $\dfrac{5}{18}$.

Because, 1 km = 1000 m and 1 hr = 3600 sec.

$\dfrac{1000}{3600} = \dfrac{5}{18}$

Hence, option 3 is the correct option.

A car is moving with a speed of 90 km/hr. Its speed in m/sec is

1. 15 m/sec
2. 25 m/sec
3. 35 m/sec
4. 45 m/sec

Answer

Given:

Speed = 90 km/hr

To convert km/hr to m/sec, multiply by $\dfrac{5}{18}$:

$90 \text { km/hr} = \Big(90 \times \dfrac{5}{18}\Big)\text{ m/sec} \\[1em] = \Big(5 \times \dfrac{5}{1}\Big)\text{ m/sec} \\[1em] = (5 \times 5)\text{ m/sec} \\[1em] = 25 \text{ m/sec}$

Hence, option 2 is the correct option.

An athlete runs at 30 km/hr. In how much time will he run a race of 200 metres?

1. 20 sec
2. 24 sec
3. 32 sec
4. 36 sec

Answer

Given:

Speed = 30 km/hr, Distance = 200 m

Let's convert 30 km/hr to m/sec:

$30 \text { km/hr} = \Big(30 \times \dfrac{5}{18}\Big)\text{ m/sec} \\[1em] = \Big(5 \times \dfrac{5}{3}\Big)\text{ m/sec} \\[1em] = \Big(\dfrac{25}{3}\Big)\text{ m/sec}$

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

Substituting the values in above, we get:

$\text {Time} = \dfrac{200}{\dfrac{25}{3}} \text {seconds} \\[1em] = 200 \times {\dfrac{3}{25}} \text{ seconds} \\[1em] = 8 \times {\dfrac{3}{1}} \text{ seconds} \\[1em] = 8 \times 3 \text{ seconds} \\[1em] = 24 \text{ seconds}$

Hence, option 2 is the correct option.

A car is moving at a speed of 48 km/hr. How many metres will it travel in 15 minutes?

1. 8000 m
2. 10000 m
3. 12000 m
4. 16000 m

Answer

Given:

Speed = 48 km/hr, Time = 15 mins

Let's convert time to hrs:

1 hr = 60 mins

Time = 15 mins = $\dfrac{15}{60} \text{ hr} = \dfrac{1}{4} \text{ hr}$

Distance = Speed x Time

Substituting the values in above, we get:

Distance = $48 \times \dfrac{1}{4}$ km

Distance = 12 x 1 km = 12 km

Let's convert 12 km to meters:

1 km = 1000 m

Distance = 12 km = 12 x 1000 m = 12000 m

Hence, option 3 is the correct option.

A cyclist covers 175 metres in 25 seconds. What is his speed in km per hour?

1. 35 km/hr
2. 33.6 km/hr
3. 27 km/hr
4. 25.2 km/hr

Answer

Given:

Distance = 175 m, Time = 25 seconds

Speed = $\dfrac{\text{Distance}}{\text{Time}}$

Substituting the values in above, we get:

Speed = $\dfrac{175}{25}$ m/sec = 7 m/sec

Let's convert 7 m/sec to km/hr:

$7 \text{ m/sec} = 7 \times \dfrac{18}{5} \text{ km/hr} \\[1em] = \dfrac{126}{5} \text{ km/hr} \\[1em] = 25.2 \text{ km/hr}$

Hence, option 4 is the correct option.

A car covers 108 km in first two hours and 90 km in the next one hour. Its average speed is

1. 28 km/hr
2. 33 km/hr
3. 66 km/hr
4. 99 km/hr

Answer

Given:

Distances: (D1) = 108 km, (D2) = 90 km

Time: (T1) = 2 hrs, (T2) = 1 hr

Total Distance = 108 km + 90 km = 198 km

Total Time = 2 hrs + 1 hr = 3 hrs

Average Speed = $\dfrac{\text{Total Distance}}{\text{Total Time}}$

Substituting the values in above, we get:

Average Speed = $\dfrac{198}{3}$ km/hr = 66 km/hr

Hence, option 3 is the correct option.

Fill in the blanks :

(i) A speed of 81 km/hr expressed in metres/sec is ............... .

(ii) Which is greater — a speed of 50 m/sec or 50 km/hr? ............... .

(iii) If a moving body covers equal distances in equal intervals of time, its speed is said to be ............... .

(iv) In crossing a platform, a train has to cover a distance equal to the ............... of the lengths of the train and the platform.

(v) To convert a speed of m/sec into km/hr, we multiply it by ............... .

Answer

(i) A speed of 81 km/hr expressed in metres/sec is 22.5 m/sec.

(ii) Which is greater — a speed of 50 m/sec or 50 km/hr? 50 m/sec.

(iii) If a moving body covers equal distances in equal intervals of time, its speed is said to be uniform.

(iv) In crossing a platform, a train has to cover a distance equal to the sum of the lengths of the train and the platform.

(v) To convert a speed of m/sec into km/hr, we multiply it by $\dfrac{18}{5}$.

Explanation

(i)

Given:

Speed = 81 km/hr

To convert km/hr to m/sec, multiply by $\dfrac{5}{18}$.

$81 \text { km/hr}= \Big(81 \times \dfrac{5}{18}\Big)\text{ m/sec} \\[1em] = \Big(9 \times \dfrac{5}{2}\Big)\text{ m/sec} \\[1em] = \dfrac{45}{2}\text{ m/sec} \\[1em] = 22.5 \text{ m/sec}$

Speed = 22.5 m/sec

(ii)

Let's convert 50 km/hr into m/sec:
$50 \text { km/hr} = \Big(50 \times \dfrac{5}{18}\Big)\text{ m/sec} \\[1em] = \dfrac{250}{18}\text{ m/sec} \\[1em] ≈ 13.89 \text{ m/sec}$

Now compare 50 m/sec and 13.89 m/sec,

50 m/sec > 13.89 m/sec

So, 50 m/sec is greater.

(iii) The given statement is the definition of Uniform Speed (or constant speed). If the distance covered doesn't change for every second or minute that passes, the speed is consistent.

(iv) To completely clear a platform, the front of the train must enter the platform, travel the platform's length, and then the back of the train must travel the train's own length to fully exit.

(v) Since $1 \text{ m} = \dfrac{1}{1000}\text{ km}$ and $1 \text{ sec} = \dfrac{1}{3600}\text{ hr}$, the conversion factor simplifies to $\dfrac{3600}{1000} = \dfrac{18}{5}$.

Write true (T) or false (F) :

(i) A speed of 45 km/hr is the same as $12\dfrac{1}{3}$ m/s.

(ii) Average speed = $\dfrac{\text{Total distance covered}}{\text{Total time taken}}$

(iii) If a moving body covers unequal distances in equal intervals of time, its speed is said to be instantaneous.

(iv) A speed of 1 km/hr is equal to a speed of $\dfrac{5}{18}$ m/s.

(v) A 240 m long train crosses a platform double its length in 54 sec. Then, the speed of the train is 60 km.

Answer

(i) False
Reason —

To convert 45 km/hr to m/s, multiply by $\dfrac{5}{18}$.

$45 \text { km/hr} = \Big(45 \times \dfrac{5}{18}\Big)\text{ m/s} \\[1em] = \Big(5 \times \dfrac{5}{2}\Big)\text{ m/s} \\[1em] = \dfrac{25}{2}\text{ m/s} \\[1em] = 12.5 \text{ m/s}$

Let's convert $12\dfrac{1}{3}$ m/s to decimal:

$12\dfrac{1}{3}\text{ m/s} = \dfrac{37}{3}\text{ m/s} = 12.33 \text{ m/s}$

12.5 m/s is not the same as $12\dfrac{1}{3} \text{ m/s } (12.33)\text{ m/s}$.

(ii) True
Reason —

Average speed = $\dfrac{\text{Total distance covered}}{\text{Total time taken}}$

This is the standard mathematical definition for Average Speed, used to calculate the speed over a journey with multiple parts.

(iii) False
Reason — If the distances are unequal, the speed is called Non-uniform or Variable speed. "Instantaneous speed" refers to the speed at a specific moment in time (like what a car's speedometer shows).

(iv) True
Reason — To convert km/hr to m/s, multiply by $\dfrac{5}{18}$.

Speed = 1 km/hr = $\Big(1 \times \dfrac{5}{18}\Big)\text{ m/s} = \dfrac{5}{18} \text{ m/s}$

(v) False
Reason —

Train length = 240 m

Platform length = 2 x 240 = 480 m

Total Distance = 240 m + 480 m = 720 m

Time = 54 sec

Speed = $\dfrac{\text{Distance}}{\text{Time}}$

Substituting values in the above, we get:

Speed = $\dfrac{720 \text{ m}}{54 \text{ sec}} = \dfrac{40}{3} \text{ m/s}$

Let's convert speed to km/hr:

$\dfrac{40}{3} \text{ m/s} = \Big(\dfrac{40}{3} \times \dfrac{18}{5}\Big)\text{ km/hr} \\[1em] = (8 \times 6)\text{ km/hr} \\[1em] = 48 \text{ km/hr}$

Kapil is an expert cyclist. Every morning he starts from his home on his cycle and rides on the road running parallel to the railway track, to reach the railway crossing, 27 km away from his home.

(1) Today he managed to reach the railway crossing in 3 hours. Find his speed in m/sec :

1. 2 m/sec
2. 2.5 m/sec
3. 3 m/sec
4. 4.5 m/sec

(2) If tomorrow he wishes to cover this distance in 2 hours, by how much does he need to increase his speed ?

1. 1.25 m/s
2. 1.5 m/s
3. 1.75 m/s
4. 2.25 m/s

(3) Today, he has to stop at the railway crossing for a train to pass. What time will the train 180 m long, running at 54 km per hour take to pass him :

1. 9 sec
2. 10 sec
3. 12 sec
4. 15 sec

(4) What time will this train take to pass a platform 300 m long at the speed of 10 m/s?

1. 30 sec
2. 32 sec
3. 36 sec
4. 35 sec

Answer

(1)

Given:

Distance = 27 km = 27 x 1000 m = 27000 m

Time = 3 hours = 3 x 3600 sec = 10800 sec

Speed = $\dfrac{\text{Distance}}{\text{Time}}$

Substituting values in the above, we get:

Speed = $\dfrac{27000}{10800}\text{ m/sec} = \dfrac{270}{108}\text{ m/sec} = 2.5\text{ m/sec}$

Hence, option 2 is the correct option.

(2)

Given:

Distance = 27000 m

Time = 2 hours = 2 x 3600 sec = 7200 sec

Old Speed = 2.5 m/sec

New Speed = $\dfrac{\text{Distance}}{\text{Time}}$

Substituting values in the above, we get:

New Speed = $\dfrac{27000}{7200}\text{ m/sec} = \dfrac{270}{72}\text{ m/sec} = 3.75\text{ m/sec}$

Let's calculate the increase in speed:

Increase = New Speed - Old Speed

Increase = 3.75 m/sec - 2.5 m/sec = 1.25 m/sec

Hence, option 1 is the correct option.

(3)

Given:

Distance = Train length = 180 m

Speed = 54 km/hr

Let's convert speed to m/sec:

$54 \text { km/hr} = \Big(54 \times \dfrac{5}{18}\Big)\text{ m/sec} \\[1em] = \Big(3 \times \dfrac{5}{1}\Big)\text{ m/sec} \\[1em] = 15 \text{ m/sec}$

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

Substituting the values in above, we get:

Time = $\dfrac{180}{15}\text{ sec} = 12 \text{ sec}$

Hence, option 3 is the correct option.

(4)

Given:

Train length = 180 m

Platform length = 300 m

Total Distance = 180 m + 300 m = 480 m

Speed = 10 m/sec

Time = $\dfrac{\text{Total Distance}}{\text{Speed}}$

Substituting values in the above, we get:

Time = $\dfrac{480}{10}\text{ sec} = 48 \text{ sec}$

Time taken by the train to pass the platform is 48 sec.

Therefore, none of the provided options is correct for the given values.

Assertion: A car is travelling at a speed of 48 km/h. In 35 minutes, it will cover 28 km.

Reason: Distance = speed x time

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

Given:

Speed = 48 km/hr

Distance = 28 km

Time = 35 minutes

Let's convert time to hours:

1 hour = 60 minutes

Time = 35 minutes = $\dfrac{35}{60}\text{ hr} = \dfrac{7}{12}\text{ hr}$

Distance = Speed x Time

Substituting the values in above, we get:

Distance = $\Big(48 \times \dfrac{7}{12}\Big)\text{ km} = \Big(4 \times \dfrac{7}{1}\Big)\text{ km} = 28 \text{ km}$

The Assertion is True.

Reason:

Distance = speed x time

This is the correct, standard formula for distance.

Hence, option 1 is the correct option.

Assertion: Vivek goes from his village to a city at 6 km/h and returns back at a speed of 4 km/h. If the distance between village and the city is 12 km, then his average speed for the whole journey is 4.8 km/h.

Reason: Average speed = $\dfrac{\text{Onward speed + return speed}}{2}$

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is true but Reason (R) is false.

Explanation

Given:

Distance = 12 km

Onward speed = 6 km/hr

Return speed = 4 km/hr

Let's find total time:

Onward time (T1) = $\dfrac{\text{Distance}}{\text{Onward speed}} = \dfrac{12}{6}\text{ hrs} = 2 \text{ hrs}$

Return time (T2) = $\dfrac{\text{Distance}}{\text{Return speed}} = \dfrac{12}{4}\text{ hrs} = 3 \text{ hrs}$

Total Time = 2 hrs + 3 hrs = 5 hrs

Total Distance = 12 km (to city) + 12 km (back home) = 24 km

Average speed = $\dfrac{\text{Total Distance}}{\text{Total Time}}$

Substituting the values in above, we get:

Average speed = $\dfrac{24}{5}\text{ km/hr} = 4.8 \text{ km/hr}$.

The Assertion is True.

Reason:

Average speed = $\dfrac{\text{Onward speed + return speed}}{2}$

This is False. You cannot simply find the average of the speeds. You must always use $\dfrac{\text{Total Distance}}{\text{Total Time}}$.

The reason is false.

Hence, option 3 is the correct option.

If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. The actual distance travelled by him is:

1. 50 km
2. 56 km
3. 70 km
4. 80 km

Answer

Let the actual distance travelled by the person be d km.

Let the actual time taken be t hours.

Case 1 (Actual): Speed = 10 km/hr, Distance = d km

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

Substituting the values in above, we get:

t = $\dfrac{d}{10}$ hours ........(i)

Case 2 (New): Speed = 14 km/hr, Distance = (d + 20) km

In the same time t,

Distance = Speed × Time

Substituting the values, we get:

d + 20 = 14 × t

d + 20 = 14 × $\dfrac{d}{10} \quad$ [From (i)]

d + 20 = $\dfrac{14d}{10}$

d + 20 = $\dfrac{7d}{5}$

5(d + 20) = 7d

5d + 100 = 7d

⇒ 7d - 5d = 100

⇒ 2d = 100

⇒ d = 50 km

Hence, option 1 is the correct option.

Sound waves travel at 300 m/s. Sound produced at a point is heard by a person after 5 seconds, while the same sound is heard by another person after 6 seconds. What could be the maximum and minimum distance between the two persons?

1. 1.8 km, 0.15 km
2. 2.2 km, 0.20 km
3. 2.8 km, 0.25 km
4. 3.3 km, 0.3 km

Answer

Given:

Speed of sound = 300 m/s

Time taken by person 1 to hear the sound (T₁) = 5 seconds

Time taken by person 2 to hear the sound (T₂) = 6 seconds

Distance = Speed x Time

Let's find the distance of each person from the source of the sound:

Distance of person 1 from source (D₁) = 300 x 5 = 1500 m = 1.5 km

Distance of person 2 from source (D₂) = 300 x 6 = 1800 m = 1.8 km

Maximum distance occurs when the two persons are on opposite sides of the source.

Maximum distance = D₁ + D₂ = 1.5 km + 1.8 km = 3.3 km

Minimum distance occurs when the two persons are on the same side of the source.

Minimum distance = D₂ - D₁ = 1.8 km - 1.5 km = 0.3 km

Hence, option 4 is the correct option.

A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 11 km/hr respectively. The distance between them after 6 minutes is:

1. 100 m
2. 150 m
3. 190 m
4. 200 m

Answer

Given:

Initial distance between thief and policeman = 200 m

Speed of thief = 10 km/hr

Speed of policeman = 11 km/hr

Time = 6 minutes

Since both are running in the same direction, the policeman gains on the thief at the relative speed.

Relative speed = Speed of policeman - Speed of thief

Relative speed = 11 km/hr - 10 km/hr = 1 km/hr

Let's convert relative speed to m/min:

1 km/hr = $\dfrac{1000 \text{ m}}{60 \text{ min}} = \dfrac{50}{3}$ m/min

Distance gained by policeman in 6 minutes

Distance = Speed x Time

Substituting the values, we get:

Distance gained = $\Big(\dfrac{50}{3} \times 6\Big)\text{ m} = \dfrac{300}{3}\text{ m} = 100 \text{ m}$

Remaining distance between them = Initial distance - Distance gained

Remaining distance = 200 m - 100 m = 100 m

Hence, option 1 is the correct option.

A car covers 108 km in the first two hours and 90 km in the next one hour. Its average speed is:

1. 28 km/hr
2. 33 km/hr
3. 66 km/hr
4. 99 km/hr

Answer

Given:

Distance: (D₁) = 108 km, (D₂) = 90 km

Time: (T₁) = 2 hrs, (T₂) = 1 hr

Total Distance = 108 km + 90 km = 198 km

Total Time = 2 hrs + 1 hr = 3 hrs

Average Speed = $\dfrac{\text{Total Distance}}{\text{Total Time}}$

Substituting the values in above, we get:

Average Speed = $\dfrac{198}{3}$ km/hr = 66 km/hr

Hence, option 3 is the correct option.

A man decided to cover a distance of 6 km in 84 minutes. He decided to cover two-thirds of the distance at 4 km/hr and the remaining at some different speed. His speed after the two-thirds distance has been covered is:

1. 5 km/h
2. 7 km/h
3. 9 km/h
4. 3 km/h

Answer

Given:

Total Distance = 6 km

Total Time = 84 minutes

Let's convert total time to hours:

1 hour = 60 minutes

∴ 84 minutes = $\dfrac{84}{60}\text{ hr} = \dfrac{7}{5}\text{ hr}$

Distance covered in first part = $\dfrac{2}{3} \times 6$ km = 4 km

Speed for first part = 4 km/hr

Time taken for first part (T₁) = $\dfrac{\text{Distance}}{\text{Speed}}$

Substituting the values in above, we get:

T₁ = $\dfrac{4}{4}$ hr = 1 hr

Remaining distance = 6 km - 4 km = 2 km

Remaining time (T₂) = Total Time - T₁

T₂ = $\dfrac{7}{5}$ hr - 1 hr = $\dfrac{7 - 5}{5}$ hr = $\dfrac{2}{5}$ hr

Speed for remaining distance = $\dfrac{\text{Remaining Distance}}{\text{Remaining Time}}$

Substituting the values in above, we get:

$\text{Speed} = \dfrac{2}{\dfrac{2}{5}}\text{ km/hr} \\[1em] = \Big(2 \times \dfrac{5}{2}\Big)\text{ km/hr} \\[1em] = 5 \text{ km/hr}$

Hence, option 1 is the correct option.