Assertion (A): If momentum of a body increases by 50%, its kinetic energy will increase by 125%.

Reason (R): Kinetic energy is proportional to the square of velocity.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true and R is not the correct explanation of A.
3. assertion is false but reason is true.
4. assertion is true but reason is false.

Both A and R are true and R is the correct explanation of A.

Explanation

Assertion (A) is true. When the momentum of a body increases by 50%, its kinetic energy will increase by 125%.

Let initial momentum be p and kinetic energy be K

If p increases by 50%, the new momentum p' = p + $\dfrac{p}{2}$ = $\dfrac{3p}{2}$

Relation between kinetic energy and momentum is given by; K = $\dfrac{p^2}{2m}$

So, $\dfrac{K'}{K}$ = $\dfrac{\dfrac{p'^2}{2m}}{\dfrac{p^2}{2m}}$ = $\dfrac{p'^2}{p^2}$ = $\dfrac{\Big(\dfrac{3p}{2}\Big)^2}{\dfrac{p^2}{1}}$ = $\dfrac{9}{4}$

So,

Percentage increase in kinetic energy = $\dfrac{K'-K}{K}$ x 100

= $\dfrac{ \dfrac{9K}{4} -K}{K}$ x 100

= $\dfrac{ \dfrac{9K-4K}{4}}{K}$ x 100

= $\dfrac{ \dfrac{5K}{4}}{K}$ x 100

= $\dfrac{5}{4}$ x 100 = 125%

Hence, kinetic energy will increase by 125%.

Reason (R) is true. Kinetic energy is proportional to the square of velocity, according to the formula: KE = $\dfrac{1}{2}$mv².