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The ratio of acceleration due to gravity...

The ratio of acceleration 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?

A

`5:2`

B

`sqrt(5):sqrt(2)`

C

`5:2sqrt(2)`

D

`25:4`

Text Solution

Verified by Experts

The correct Answer is:
C
`g_1:g_2=5:2`……….(1)
and `rho_1:rho_2=2:1` ………(2)
Escape velocity ,`v=sqrt(2gR)`
From (1) ,`((GM_1)/(R_1^2))/((GM_2)/(R_2^2))=5/2 or, (M_1R_2^2)/(M_2R_1^2)=5/2 or, (M_1)/(M_2)=(5R_1^2)/(2R_2^2)`
From (2) ,
`((M_1)/(4/3piR_1^3))/((M_2)/(4/3piR_2^3))=2/1 or, (M_1xxR_2^3)/(M_2xxR_1^3)=2`
`or, (R_2^3)/(R_1^3)=2xx(M_2)/(M_1) or,(R_2^3)/(R_1^3)=2xx(2R_2^2)/(5R_1^2)or, (R_2)/(R_1)=4/5`
`therefore (v_1)/(v_2)=sqrt((R_1)/(R_2)xx(g_1)/(g_2))=sqrt(5/4xx5/2)=5/(2sqrt(2))`
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