Question

How do you find the instantaneous velocity of the particle?

Answer 1

Your question is too vague. You must have a given, that is, you should be given a function for displacement or acceleration.

If you are given a displacement function, then you simply need to find the derivative to get the velocity function. If you are given the basic kinematic formula:
`d(t)=1/2at^2+v_it`,
you can find the derivative wrt `t` using the power rule. So:
`d'(t)=v(t)=at+v_i`

If you are given an acceleration function, you will also require some known velocity at a known time to get a particular solution, otherwise you just get a general solution. Given an acceleration function, you need to integrate to get the velocity function. If you are given the basic kinematic formula:
`a(t)=a`, that is a constant acceleration function.
you can find the integral wrt `t` using the power rule. So:
`v(t)=int a dt=at+C`
If you are given `v_i` at `t=0`, then you get:
`v(t)=at+v_i`

As you can see with a constant acceleration problem, we can get the velocity function from either the displacement function or the acceleration function and the result is the same.

With calculus, you can find the velocity function from more complicated displacement and acceleration functions and you won't get bored sticking the the basic kinematic functions!