Why are the equations of motion valid only when acceleration is constant? Also, why are- dv/dt , ds/dt etc valid in all cases? Don't we consider acceleration constant while deriving them?

Answer 1

When the equations of motion are derived using constant values of acceleration, we get the equation you are used to seeing. However, in the more general case, where acceleration is some function of time (and not just a number/constant, then it still equals dv/dt,

Example: If dv/dt = 5 (acceleration is a constant)

dv/dt = 5

dv = 5dt

∫dv = ∫5 dt

v = 5t + C

To find "C", for the initial condition, we set time equal to zero so C = v_i

The equivalent equation of motion is v_f = at + v_i or v_f = v_i + at

Then, starting with that and using ds/dt = v:

ds/dt = 5t + v_i

ds = 5t + v_i dt

∫ds = ∫5t + v_i dt

s = 5/2t² + v_it + C

s = (1/2)(5)t² + v_it + C

Solving for "C", we make t equal to zero and get C = s_i (the initial position of the object in question) which matches the equation of motion x = (1/2)at² + v_it + x_i

If acceleration were a function instead of a constant, we do the same thing but you get a more complicated equation. So the kinematic equations of motion are just a special case when acceleration is constant.