# Finding the initial velocity using calculus

I throw a stone at 20 degree, when the stone falls to the ground, it reaches 100m further. Using CALCULUS methods, find the initial velocity of the stone.

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What have you tried? –  apnorton Jan 17 '13 at 3:20
i got 39.05 as final answer with physics formula, but dunno how to use calculus to solve this :/ –  user42624 Jan 17 '13 at 3:44
The answer is correct, under the usual assumptions (airless planet with same acceleration due to gravity as Earth). For calculus solution, need some Physics concepts, such as vertical, horizontal component of velocity. –  André Nicolas Jan 17 '13 at 4:03
by the way, I'm just wandering, what will the graph of velocity as a function of time look like? –  user42624 Jan 17 '13 at 4:07

The calculus part in this is taking the derivative of $\vec{r}(t)$,which is $\dot{\vec{r}}(t)=\vec{v}(t)$ (the dot above r means derivative with respect to $t$). Most of the used concepts in this example are from physics, though.
Somewhere in your calculation, you must have $x(t)$ and $y(t)$, which represent the position of the stone dependent on time. Assume that the initial time when you throw the stone is $t=0$. Because $\vec{r}(t)=(x(t),y(t))$, $\vec{v}(t)=\dot{\vec{r}}(t)=(\dot x(t),\dot y(t))$, and thus $\vec{v}(0)=(\dot x(0),\dot y(0))$.