Escape velocities

Classical Mechanics Level 3

Two spherical planets $P$ and $Q$ have the same uniform density $\rho$ , masses $M_\rho$ and $M_{Q}$ , and surface areas $A$ and $4A$ , respectively. A spherical planet $R$ also has uniform density $\rho$ and its mass is $( M_P + M_Q)$ . The escape velocities from the planets $P, Q$ and $R$ , are $V_P, V_Q$ and $V_R$ , respectively. Then

(A) $V_Q > V_R > V_P$

(B) $V_R > V_Q > V_P$

(D) $V_P/V_Q = \frac{1}{2}$

A only B and D C only A and C

1 solution

Alex Li
Apr 24, 2015

The escape velocity on a spherical planet is given by $\sqrt{\frac{2GM}{r}}$ . Because $Q$ 's surface area is $4$ times that of $P$ , it must have twice the radius. Then, the volume is increased by a factor of $2^3=8$ . This implies that $M_Q=8M_P$ , because the densities of the planets are the same. The escape velocity of $Q$ is thus $\sqrt{\frac{2G\times8M_P}{2r_P}}=2\sqrt{\frac{2GM_P}{r_P}}=2V_P$ which satisfies condition $(D)$ . Note that only one of the answers choices contains D, so this must be the answer.

Escape velocities

1 solution

0 pending reports