# Can a body have a velocity equal to zero but non-zero acceleration?

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Let us take displacement at time t to be represented as d = f (t), where f (t) is a function of time.

Now velocity is given by the change in displacement in unit time. Instantaneous velocity is given by the derivative of the function defining displacement or f’ (t).

Acceleration is the change in velocity in unit time. Instantaneous acceleration is given by the derivative of the function defining velocity or f’’ (t).

Now if velocity is zero, f’ (t) = 0 which implies that f’’ (t) is also zero.

It is possible though for f’ (t) to be zero but f’’ (t) to have a non-zero value, but this can occur only at a few selected instances of time.

For example if d= f (t) = t^2 + t, f’ (t) = 2t and f’’ (t) = 2. At t = 0, f’ (t) = 0 but f’’ (t) is not zero. Therefore we see that the velocity at time t=0 is zero but acceleration is non-zero.

**The condition though cannot be true consistently. It is true only at one or a few particular instances of time.**

When velocity is zero, that means the body is stopped, at least for a moment. Acceleration is the rate at which velocity changes. So the question is, can the velocity be changing at the moment it is stopped? Think about tossing a ball straight up in the air. When the ball is at the peak of the toss, for that split second, it is at a velocity of zero. But the very next moment, it is moving downward. So while it was stopped, it was still accelerating. The situation you describe happens whenever something is turning around.