Finding the initial velocity using calculus

Question

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.

Answer 1

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 $\vec{v}(0)=(\dot x(0),\dot y(0))$.

The last formula is the "calculus" part of the calculation.