# How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero?

I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation.

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Suppose you throw a ball upwards at some velocity $v$. When you catch it again it's traveling downwards at (ignoring air resistance) a velocity of $-v$. So somewhere in between throwing and catching the ball it must have been stationary for a moment i.e. it's instaneous velocity was zero. Obviously this was at the top of it's travel.

When you throw the ball it immediately starts being accelerated downwards by the Earth's gravity, so it has a constant acceleration downwards of $-9.81ms^{-2}$ (the acceleration is negaitive because it's reducing the velocity of the ball).

So this is an example of how there can be a non-zero acceleration (of $-9.81ms^{-2}$) but there can be a moment when the ball's instantaneous velocity is zero.

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 Ah. Awesome, thanks! – anjunatl Aug 14 '12 at 17:22

I think it is not trivial fact. There is an interesting story about it in the history of science. Galileo Galilei spent a lot of time to understand how a body can start its motion with zero velocity. For his time it sounded like an absurd and nobody believed him. In his book «Dialogues Concerning Two New Sciences», he convinced himself (and everybody else) by the following remarkable explanations why $v(t)$ can be zero for $t=0$:
The second funny story was about equation of motion: since $v=0$ in the beginning of motion, Galilei assumed that the speed of free falling body should be proportional to the distance passed: $v(t)=a\,l(t)$, where $a$ is a some constant. Later he proved that such motion is simply impossible for the initial condition $l(0)=0$. Then he found the correct equation $v(t)=a\,t$.
Later John Napier consider Galileo's equation $v(t)=a\,l(t)$ for $l(0)\neq 0$ and thus he discover logarithmic function!