For a planet of mean density and radius

PHYSICS · Single correct · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

For a planet of mean density

\rho

and radius

R

, a student proposes that its escape velocity

v_{e}

can be written as

v_{e} = k\,(G \rho)^{m} R^{n}

, where

G

is the universal gravitational constant and

k

is dimensionless. What values of

m

and

n

make the formula dimensionally consistent? (a)

m = \dfrac{1}{2},\; n = 2

(b)

m = 1,\; n = \dfrac{1}{2}

(c)

m = \dfrac{1}{2},\; n = 1

(d)

m = \dfrac{1}{2},\; n = \dfrac{1}{2}