In summary: On Planet X, the astronaut weighs twice as much as on Earth due to the stronger gravitational force. This means that in order for the rocket to escape the gravitational pull of Planet X, it needs to have the same kinetic energy as on Earth, but with twice the weight. Since kinetic energy is directly proportional to mass, the escape velocity on Planet X remains the same as on Earth, which is represented by the letter v. Therefore, the answer is C.
  • #1
15ongm
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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?
 
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  • #2
15ongm said:
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.
 
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  • #3
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 Mplanet/Mearth. That follows from the factor 2 in g and the expression for g PeroK is asking for
 
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Related to How Does Planet X's Gravity Affect Rocket Escape Velocity?

What is escape velocity?

Escape velocity is the minimum speed needed for an object to escape the gravitational pull of a celestial body, such as a planet or moon.

How is escape velocity calculated?

Escape velocity can be calculated using the formula Ve = √(2GM/R), where Ve is the escape velocity, G is the gravitational constant, M is the mass of the celestial body, and R is the distance from the center of the body to the object.

Why is escape velocity important?

Escape velocity is important for space exploration as it determines the amount of energy needed for a spacecraft to leave the gravitational pull of a celestial body and continue on its intended trajectory.

Can escape velocity be exceeded?

Yes, escape velocity can be exceeded. This is known as hyperbolic excess velocity and occurs when an object has more energy than is needed to escape the gravitational pull of a celestial body.

How does escape velocity differ on different celestial bodies?

Escape velocity varies depending on the mass and size of a celestial body. For example, the escape velocity on Earth is 11.2 km/s, while the escape velocity on the Moon is only 2.4 km/s due to its lower mass and weaker gravitational pull.

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