91Ó°ÊÓ

Gravitational constant $G=6.67×{10}^{-11}{m}^3/kg{s}^2$

mass of the earth is role="math" localid="1649961110336" $m_e=5.98×{10}^{24}kg$

mass of the moon is role="math" localid="1649961080261" $m_m=7.36×{10}^{22}kg$

mass of the boulder is $m_b$

distance between earth and moon is role="math" localid="1649961173396" $r_{em}=3.84×{10}^8$

distance reached by the boulder from the center of the moon is $r$

The minimum speed with which a boulder must be launched to reach the earth.

Expression for System's potential energy

$U_G=-G\left(\frac{m_e{m}_b}{r_{em}-r}+\frac{m_m{m}_b}{r}+\frac{m_e{m}_m}{r_{em}}\right)$

After finding the first derivative of $U_{G,}\left(\frac{∂U_G}{∂r}\right)=0$

Equation to find$r$becomes

$$\begin{gathered} \frac{r_{em}}{r}=1Â\pm \sqrt{\frac{m_e}{m_m}} \\ ⇒\frac{3.84×{10}^8}{r}=1Â\pm \sqrt{\frac{5.98×{10}^{24}}{7.36×{10}^{22}}} \\ r=3.836×{10}^{7}m \end{gathered}$$Note: take the positive sign.

Equating the sum of initial and final potential and kinetic energies of the system

$U_i+K_i=U_f+K_f$

Since final velocity is zero, $K_f=0$

$∴K_i=-U_i+U_f$

Launch velocity$v$ is given by

role="math" localid="1649960963111" $$\begin{gathered} v=\sqrt{2G{m}_e\left(\frac{1}{r_{em}-R_m}-\frac{1}{r_{em}-r}\right)+2G{m}_m\left(\frac{1}{R_m}-\frac{1}{r}\right)} \\ v=\sqrt{2(6.67×{10}^{-11})(5.98×{10}^{24})\left(\frac{1}{3.84×{10}^8-1.74×{10}^6}-\frac{1}{3.84×{10}^8-3.836×{10}^7}\right)+2(6.67×{10}^{-11})(7.36×{10}^{22})\left(\frac{1}{1.74×{10}^6}-\frac{1}{3.836×{10}^7}\right)} \\ v=\sqrt{-2.21×{10}^5+53.87×{10}^5} \\ v=2.3×{10}^{3}m/s=2.3km/s \end{gathered}$$

Impact speed on earth of a boulder launched at minimum speed $v_f$

$$\begin{gathered} v_f=\sqrt{2G\left[m_e\left(\frac{1}{R_e}-\frac{1}{r_{em}-r}\right)+m_m\left(\frac{1}{r_{em}-R_e}-\frac{1}{r}\right)\right]} \\ {v}_f=\sqrt{2×6.67×{10}^{-11}\left[5.98×{10}^{24}\left(\frac{1}{1.74×{10}^6}-\frac{1}{3.84×{10}^8-3.836×{10}^7}\right)+7.36×{10}^{22}\left(\frac{1}{3.84×{10}^8-1.74×{10}^6}-\frac{1}{3.836×{10}^7}\right)\right]} \\ {v}_f=\sqrt{123.1×{10}^6}=11×{10}^{3}m/s=11km/s \end{gathered}$$

The impact speed on earth when boulder is launched at minimum speed$11km/s$