We know that acceleration = dv/dt.  If an object is thrown upward, v=0 at the highest point. Hence acceleration (from above relation) must be 0, but it is not since acceleration due to gravity is acting on it continuously. Hence this becomes a contradictory case. Please explain this to me.

Expert Answers

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Hello!

Luckily, there is no contradiction. You are right that `a = (dv)/dt,` that `v = 0` at the highest point and that the acceleration due to gravity is constant.

The only incorrect statement you use is that `(dv)/dt = 0` if `v=0.` It is not true in general.

Recall the definition of a derivative `(dv)/dt` or `v',` which are the same. For a function `v(t)`:

`v'(t) = lim_(Delta t->0) (v(t+Delta t) - v(t))/(Delta t).`

If `v(t) = 0` for some specific moment `t,` `v'(t)` becomes:

`v'(t) = lim_(Delta t->0) (v(t+Delta t))/(Delta t).`

But it doesn't have to be zero! For example, in our case `v(t) = V_0-g*t` is a linear function. For any `t,` even for `t = V_0/g,` where `v(t)=0,`

`(v(t+Delta t) - v(t))/(Delta T) = ((V_0-g*(t+Delta t))-(V_0-g*t))/(Delta t) = -g,`

so the limit is also `-g` (this means `+g` downwards).

There is a grain of truth in your assumption, nevertheless. If `v(t)=0` at some entire interval around a point `t_0,` then `(v(t_0+Delta t) - v(t_0))/(Delta t) = 0` for all `Delta t` small enough, and `v'(t_0)=0.`

 

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