Show that the rate of change of momentum = mass x acceleration. Under what condition does this relation hold?

When a force is applied on a moving body, it's velocity changes. Due to change in velocity of the motion, it's momentum also changes.

Let a force (F) be applied on a body having mass (m) for time (t) due to which it's velocity changes from u to v.

Then,

Initial momentum of the body = mu
Final momentum of the body = mv

Change in momentum of the body in (t) seconds

= mv - mu = m (v - u)

Then, rate of change of momentum

= $\dfrac{\text {Change in velocity}}{\text {Time}}$

= $\dfrac{\text {m(v - u)}}{\text {t}}$

And we know,

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

Therefore,

Rate of change of momentum = ma = mass x acceleration.

This relation holds true when mass of the body remains constant.