Two spherical planets P and Q have the same uniform density , masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density and its mass is (MP + MQ). The escape velocities from the planets P, Q and R, are VP, VQ and VR, respectively. Then

Engineering

Physics

Escape and Orbital Velocity

Question

Two spherical planets P and Q have the same uniform density $ρ$ , masses M_P and M_Q, and surface areas A and 4A, respectively. A spherical planet R also has uniform density $ρ$ 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

V_Q > V_R > V_P

$V_{P} / V_{Q} = \frac{1}{2}$

V_R > V_Q > V_P

V_R / V_P = 3

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$\frac{P}{ρ}$ $\frac{Q}{ρ}$ $\frac{R}{ρ}$

M 8M 9M

A 4A

R 2R 3^2/3 R

$V_{P} = \sqrt{\frac{GM}{R}}$

$V_{Q} = \sqrt{\frac{G 8 M}{2 R}} = 2 \sqrt{\frac{GM}{R}} = 8^{1 / 3} \sqrt{\frac{GM}{R}}$

$∴ \frac{V_{P}}{V_{Q}} = \frac{1}{2}$

$V_{R} = \sqrt{\frac{G (9 M)}{3^{2 / 3} R}} = \sqrt{3^{4 / 3} \frac{GM}{R}} = 3^{2 / 3} \sqrt{\frac{GM}{R}} = 9^{1 / 3} \sqrt{\frac{GM}{R}}$

$∴$ V_R > V_Q > V_P