Consider a planet that may be treated as a uniform sphere of radius and mean density .

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

Consider a planet that may be treated as a uniform sphere of radius

R

and mean density

\rho

. (i) Express the mass

M

of the planet in terms of

R

\rho

, and fundamental constants. (ii) Using energy considerations, derive an expression for the escape velocity from the planet’s surface in terms of

G

\rho

, and

R

. (iii) Hence show that

v_{\text{esc}} \propto R\sqrt{\rho}

. (iv) Planet

A

has

R_A = 5.0\times10^{6}\,\text{m}

and

\rho_A = 4500\,\text{kg\,m}^{-3}

, while planet

B

has

R_B = 7.0\times10^{6}\,\text{m}

and

\rho_B = 3200\,\text{kg\,m}^{-3}

. Calculate the ratio

v_{\text{esc},B}/v_{\text{esc},A}

. Take

G = 6.67\times10^{-11}\,\text{N\,m}^2\text{kg}^{-2}