Question

The ratio of escape velocity at earth `(v_(e))` to the escape velocity at a planet `(v_(y))` whose radius and density are twice
A. `1 : 2`
B. `1 : 2sqrt(2)`
C. `1 : 4`
D. `1 : sqrt(2)`

Answer 1

Correct Answer - B
On earth, `g_(e) = (GM)/(R_(e)^(2)) = (G)/(R_(e)^(2)) xx (4)/(3) piR_(e)^(2)rho_(e)`
`= (4)/(3) pi GR_(e) rho_(e)`
Escape velocity, `upsilon_(e) = sqrt(2g_(e)R_(e)) = R_(e) sqrt((8)/(3) piG rho_(e))`
On planet, `upsilon_(p) = R_(p)sqrt((8)/(3) pi G rho_(p))`
`:. (upsilon_(e))/(upsilon_(p)) = (R_(e))/(R_(p)) sqrt((rho_(e))/(rho_(e))) = (R_(e))/(2R_(e)) sqrt((rho_(e))/(2rho_(e))) = (1)/(2sqrt(2))`