Energy required to go from one planet to another

[addy899]

1. Description of Problem
A binary planet system comprises two identical planets of mass M and radius R with their centers a distance 10 R apart. The minimum energy that the engines of a spacecraft need to supply to get a rocket of mass m from the surface of one planet to the surface of the other is of the form XGMm/R. What is X?

Homework Equations

Escape Velocity v= √2GM/R
Gravitational Kinetic Energy KE=GMM/2R
Gravitational Potential Energy U = GmM/R
Kinetic Energy = 1/2mv²

3. Attempt at a solution

use escape velocity to find the KE of escape velocity is KE = mMG/R

I'm not sure where to go from here...

subtract the potential energy that the second planet supplies once the rocket reaches halfway?

this gives x=.8

 

[mfb]

I think you are supposed to make some approximations, otherwise the problem becomes really complicated.
- neglect orbital mechanics and how rockets work
- neglect the motion of the planets around each other?

If you just consider the potential energy at the surface of a planet and at the center between the planets, you get some nice answer (it is not 0.8, however - don't forget the potential of the other planet).

 

[addy899]

What we need to find is the amount of energy to get the rocket half way to the other planet, right? this can be found y finding the difference between the potential energies at the surface and half way through?

Potential at the surface is MmG/R - MmG/9R = 8MmG/9R
The second term is the Potential E from the second planet - I didn't include this in my previous attempt

potential halfway is MmG/5R

this gives X=.688

This is close to the answer which is listed as X=0.7

Is this correct?

 

[barryj]

I understand this "Potential at the surface is MmG/R - MmG/9R = 8MmG/9R"

How did you get this "potential halfway is MmG/5R"

 

[addy899]

Halfway between the center of the two planets is 5R, so the potential E from the second planet at half way is MmG/5R

Conceptually, the potential at that point is 0 because the planets pull on it equally?

 

[mfb]

Halfway between the center of the two planets is 5R, so the potential E from the second planet at half way is MmG/5R

And the potential from the other planet?

Conceptually, the potential at that point is 0 because the planets pull on it equally?

The force is 0, the potential is not (with your definition of the potential).

 

[addy899]

the potential from the other planet is the same...

so the potential at the midpoint is 2MmG/5R?

 

[mfb]

I think there is a minus sign missing everywhere, but apart from that: right.

 

[barryj]

How can there be a potential if there is no force?
Also, consider one of your previous equations,.. Potential at the surface is MmG/R - MmG/9R = 8MmG/9R ,
using this, the potential halfway between would be zero because the first term would change to MmG/5R as well as the second term and the difference would be zero it seems.

 

[mfb]

How can there be a potential if there is no force?

There is nothing wrong with that. Actually, every potential minimum, maximum* and critical point has no (net) force, while every point in space has a potential.

*does not exist in gravity

Potential at the surface is MmG/R - MmG/9R = 8MmG/9R

There is a sign error, too.

 

[barryj]

If you do the calculus and integrate the f dl from R to 5R you get 8MnG/9R, so I think this would be the correct answer. In a similar manner if you integrate from R to infinity you will get 0,8, not 8/9. I am a bit confused but will figure this out eventually, unless someone shows me the way.