How Does Planet X's Gravity Affect Rocket Escape Velocity?

[15ongm]

1. Problem
A rocket has landed on Planet X, which has half the radius of Earth. An astronaut onboard the rocket weighs twice as much on Planet X as on Earth. If the escape velocity for the rocket taking off from Earth is v , then its escape velocity on Planet X is

a) 2 v
b) (√2)v
c) v
d) v/2
e) v/4

The answer is C.

I reasoned out this problem mathematically, but what is the conceptual reasoning behind this?

 

[PeroK]

1. Problem
A rocket has landed on Planet X, which has half the radius of Earth. An astronaut onboard the rocket weighs twice as much on Planet X as on Earth. If the escape velocity for the rocket taking off from Earth is v , then its escape velocity on Planet X is

a) 2 v
b) (√2)v
c) v
d) v/2
e) v/4

The answer is C.

I reasoned out this problem mathematically, but what is the conceptual reasoning behind this?

I would say this is a mathematical question. To see it more clearly, you could try to derive a formula for escape velocity in terms of the radius ##R## and the surface gravity ##g## of a planet.

 

[BvU]

Conceptually you are supposed to quickly approach this from the other end: ##{\tfrac 1 2}mv_{esc}^2## is the kinetic energy needed to overcome the gravitational potential energy. I expect you know how to write the latter as ##GM_{planet}\over R_{planet}##. To see how this ratio relates to idem Earth you need M_planet/M_earth. That follows from the factor 2 in g and the expression for g PeroK is asking for