A reference book misprints the escape speed from the surface of a planet of mass and…

PHYSICS · Multi-part · CBSE · Physics · Units and Measurements · Units and Measurements

Problem

A reference book misprints the escape speed

v_{e}

from the surface of a planet of mass

M

and radius

R

v_{e}=\sqrt{\dfrac{2GM}{R^{3}}},

where

G

is the gravitational constant. (i) Show, by analysing dimensions, that the printed formula is incorrect. (ii) Obtain the correct dependence of

v_{e}

G

M

, and

R

, retaining the numerical factor

K

. (iii) Using

K=\sqrt{2}

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

M_{\oplus} = 6.0\times10^{24}\,\text{kg}

, and

R_{\oplus}=6.4\times10^{6}\,\text{m}

, compute the escape speed from Earth to two significant figures.