Bell_curve

AskSN

Can you solve this physics problem?

Gravitational potential energy = $$U$$ = $$\frac{GMm}{R}$$

Escape velocity = $$v_{esc}$$ = $$\sqrt{\frac{2GM}{r}}$$

Kinetic energy of launch:

$$
\begin{align}
K_0 &= \frac{1}{2}m(2v_{esc})^2 = \frac{1}{2} m 4v_{esc}^2 \\
&= 2 m v_{esc}^2 = 2m \frac{2GM}{r} \\
&= 4 U
\end{align}
$$ 

Kinetic energy after launch = $$K_0 - U$$ = $$3U$$

$$
\begin{align}
3U &= 3 \frac{GMm}{r} = \frac{3}{2} \frac{2GM}{r} \\
&= \frac{3}{2} v_{esc}^2 = \frac{1}{2} \left(\sqrt{3}v_{esc}\right)^2
\end{align}
$$

So the answer is D) $$\sqrt{3}v_{esc}$$