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?
Best Answer
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 $a = 0$.