The ratio of accelerations due to gravity `g_(1) : g_(2)` on the surface of two planets is `5 : 2` and the ratio of their respective average densities `rho_(1) : rho_(2)` is `2 : 1`. What is the ratio of respective escape velocities `v_(1) : v_(2)` from the surface of the planets ?

Options

A. `5: 2`
B. `sqrt(5) : sqrt(2)`
C. `5 : 2sqrt(2)`
D. `25 : 4`

Answer

C

Explanation

Escape velocity of an object from the surface of the planet is given by
`v_(e )=sqrt((2GM)/(R ))=sqrt(2gR)" "` ….(i)
when g is gravity, R is the radius of the planet, G is the gravitational constant and M is the mass of the planet. As mass`="volume" xx" density or " M=(4)/(3)piR^(3)rho`
`:. M_(1)=(4)/(3)piR_(1)^(3)rho_(1)" and " M_(2)=(4)/(3)piR_(2)^(3)rho_(2)`
Let `M_(1)` and `M_(2)` be the masses of the planets and `rho_(1)` and `rho_(2)` be the densities of the planets.
As we know, `g=(GM)/(R^(2))`
`:. (g_(1))/(g_(2))=(M_(1)R_(2)^(2))/(M_(2)R_(1)^(2))=(5)/(2) rArr ((4)/(3)piR_(1)^(3)xxR_(2)^(2)rho_(1))/((4)/(3)piR_(2)^(3)xxR_(1)^(2)rho_(2))=(5)/(2)`
`[:. g_(1) : g_(2)=5 : 2]`
`" or (R_(1))/(R_(2))=(5)/(2)xx(rho_(2))/(rho_(1))=(5)/(2)xx(1)/(2)=(5)/(4)[`:.` rho_(1) : rho_(2)=2 : 1]`
Now, the ratio of respective escape velocities
`v_(e_(1)) : v_(e_(2))=sqrt(g_(1)R_(1)):sqrt(g_(2)R_(2))=(5)/(2sqrt(2))" "`[Using eqn.(i)]