A body of mass m is thrown at angle α to the horizontal with…

A body of mass $m$ is thrown at angle $\alpha$ to the horizontal with speed $v_0$. Find the instantaneous power of the gravitational force as a function of time.

Explanation of Method

Power of gravity on a projectile.

Step 1: Power of gravity

\[ P = \vec F \cdot \vec v = -mg\,v_y = -mg\,(v_0\sin\alpha - gt). \]

\[P(t) = mg\,(gt - v_0\sin\alpha)\]

It is negative while the body rises, zero at the top ($t = v_0\sin\alpha/g$) and positive while it falls.

$P$ $t$ $-mgv_0\sin\alpha$ $t = \tfrac{v_0\sin\alpha}{g}$ $\text{rising: } P < 0$ $\text{falling: } P > 0$ $P = mg\,(gt - v_0\sin\alpha)$

Power of gravity against time: negative while the body rises (gravity takes kinetic energy away), zero at the top, positive while it falls.

Result

\[\boxed{\,P(t) = mg\,(gt - v_0\sin\alpha)\,}\]

Remark

Check. The original result is correct.