If velocity for an interval is zero then is acceleration also zero throughout?

Question

We know that if velocity is zero for an instant, then acceleration need not be zero (a simple example of which is a ball thrown upwards.)

But if velocity is zero for an interval, will acceleration also be zero throughout?

I thought it should be true, as the instant $v$ comes to $0$, $a$ should become $0$ to allow the velocity to remain zero throughout that interval

Q. Is this true? How can we show this?

I tried the following:

let $a=kt$ (a is dependent on t) $$\implies \dfrac{dv}{dt} = kt$$

$$\int^v_{v_0} dv = \int^t_{t_0} kt~dt$$

$$v-v_0 = \dfrac{k(t^2 - t^2_0)}{2}$$

Now if $v_0$ to $v$ be interval when velocity is zero, $k = 0$ implying acceleration is zero ($kt=0$).

Q. Is this correct way to show this?

Answer 1

The acceleration is the time derivative of the velocity:

$$ a = \frac{dv}{dt} $$

so if the velocity does not change with time the acceleration is necessarily zero. Since in your example the velocity is constant during the interval that means $dv/dt = 0$ and therefore that $a = 0$ during the interval.

The velocity doesn't have to be zero. Any constant non-zero value will also give $a = 0$.

Answer 2

Here average velocity is zero let time be infinitesimally small then the same average velocity now become instantaneous velocity for the same interval of time. Let x and x+ dx be the displacement point in interval t to t+dt and hence let v and v+dv be the velocities at those points now instant acceleration = {v+dv} - {v} / dt which is zero.