Question

We know that we get work done upon integrating Force. But F=ma, and since m is a constant, we are left with a to integrate, which equals to velocity.
So, why dont we write W=mv
or momentum?

Answer 1

See when you integrate on both sides with respect to some quantity, you should only integrate both sides with that particular quantity itself, not with respect to 2 different quantities.

Now the mistake:- as u mentioned F=ma

U have integrated on both sides. Yes good.!! But you forgot that you are allowed to integrate only with respect to one variable. Like time or displacement. Now,

Work is function of displacement dW=F.dx, whereas ,

Velocity is the function of time.

dV=A.dt.

When you are integrating F=ma on both side, either you should integration with respect to time or with respect to displacement not both together. So now,

F=ma

Integrating on both sides with respect to time, we get,

ΔJ = mv ( where ΔJ is change in impulse)

Change in impulse is nothing but Linear momentum.

So. P=mv.

I hope you have understood the answer, if not, please comment and let me know where didn't you understand.

Thanks,

A JEE aspirant.