Question

The radius of a planet is twice the radius of earth. Both have almost equal average mass-densities. $v_P$ and $v_E$ are escape velocities of the planet and the earth, respectively, then

• A. $v_p=1.5v_E$
• B. $v_p=2v_E$
• C. $v_E=3v_p$
• D. $V_E=1.5V_p$

Answer 1

Answer: (B)

Here, $R_P=2R_E,{\rho }_E={\rho }_P$
Escape velocity of the earth,
$V_E=\sqrt{\frac{2G{M}_E}{R_E}}=\sqrt{\frac{2G}{R_E}\left(\frac{4}{3}\pi R_E^3{\rho }_E\right)}$
$=R_E\sqrt{\frac{8}{3}\pi G{\rho }_E}\dots$ (i)
Escape velocity of the planet
$V_P=\sqrt{\frac{2G{M}_p}{R_p}}=\sqrt{\frac{2G}{R_p}\left(\frac{4}{3}\pi R_p^3{\rho }_p\right)}$
$=R_p\sqrt{\frac{8}{3}\pi G_p}.$
Divide (i) by (ii), we get
$\frac{V_E}{V_P}=\frac{R_E}{R_P}\sqrt{\frac{{\rho }_E}{{\rho }_P}}$
$\frac{V_E}{V_P}=\frac{R_E}{2R_E}\sqrt{\frac{{\rho }_E}{{\rho }_E}}=\frac{1}{2}$
or $V_P=2V_E$