Initial velocity to travel between planets of same m and R

[Hashiramasenju]

Homework Statement

So there are two planets A and B whose centres are 6r apart. A and B has the same mass M and radius R. What is the minimum velocity(from the surface of B) of the rocket required to launch it from B so that it reaches A ?

Homework Equations

F=GmM/R^2

The Attempt at a Solution

So what i thought was at the mid point between the planets the rocket must have zero velocity so
acceleration=GM/x^2 -GM/(6r-x)^2
and as a=v dv/dx
then i integrated both sides with limits of 0&Vm(launch speed) for dv
and 3R&5R for dx

Is this right ?

 

[Buzz Bloom]

Is this right ?

Hi Hashiramasenju:

I confess I find the problem as stated to be a bit confusing.
1. Do the two planets follow the same circular orbit, or are the planes of the orbits different?
2a. If the planets move in the same plane, are both moving clockwise with respect to the common axis of revolution, or are they moving in opposite directions and will eventually crash together?
2b. There is a similar question if the planets' planes of motion are different, but it is more difficult to state simply.
3a. If the planets move in the same plane, is the rocket to take off from the lagging planet or the leading planet.
3b. Similar question as 3a except for different planes of motion,
4. Assuming the same plane, is one planet assumed to be at a Legrangian point with respect to the other planet? See about "Legrangian point" here:

https://en.wikipedia.org/wiki/Lagrangian_point .​

5. If the planets are not at Legrangian point of each other, should you take into account that the planets will eventually collide due to their mutual attraction.

I think you need to make some assumptions about what the teacher intended about this problem, and include your assumptions in your answer.

About your answer: If you make a simple assumption that the two orbits are the same, then I think your solution fails to take into account that the direction of attraction between the planets in not along the orbit's path.

I hope this is some help. Good luck.

Regards,
Buzz

 

[Hashiramasenju]

Hi Hashiramasenju:

I confess I find the problem as stated to be a bit confusing.
1. Do the two planets follow the same circular orbit, or are the planes of the orbits different?
2a. If the planets move in the same plane, are both moving clockwise with respect to the common axis of revolution, or are they moving in opposite directions and will eventually crash together?
2b. There is a similar question if the planets' planes of motion are different, but it is more difficult to state simply.
3a. If the planets move in the same plane, is the rocket to take off from the lagging planet or the leading planet.
3b. Similar question as 3a except for different planes of motion,
4. Assuming the same plane, is one planet assumed to be at a Legrangian point with respect to the other planet? See about "Legrangian point" here:

https://en.wikipedia.org/wiki/Lagrangian_point .​

5. If the planets are not at Legrangian point of each other, should you take into account that the planets will eventually collide due to their mutual attraction.

I think you need to make some assumptions about what the teacher intended about this problem, and include your assumptions in your answer.

About your answer: If you make a simple assumption that the two orbits are the same, then I think your solution fails to take into account that the direction of attraction between the planets in not along the orbit's path.

I hope this is some help. Good luck.

Regards,
Buzz

Hi thanks for your reply. In this question we can ignore any motion between the planets.

 

[gneill]

So there are two planets A and B whose centres are 6r apart. A and B has the same mass M and radius R.

Is r different than R? I don't see why they would introduce another variable to the problem if it doesn't lend information about scale or symmetry.

 

[J Hann]

Have you looked at the initial and final potential energies?
What kinetic energy is required to satisfy those conditions?

 

[Buzz Bloom]

Hi thanks for your reply. In this question we can ignore any motion between the planets.

Hi @Hashiramasenju:

I think I am now getting an idea about what the problem statement means.

1. The two planets may be assumed to be stationary with respect to each and there are no other gravitational influences near by.
2. Although the planets would actually move towards each other, the problem statement assumes they do not.​

With these assumptions, your overall approach seems OK, but there seems to be something not quite right. You use both r and R. I assume that these variables are the same. That is:

3. The distance between the centers of the two planets is six time their common radius r.​

Now you have

A and B has the same mass M and radius R.

Also

What is the minimum velocity(from the surface of B) of the rocket required to launch it from B so that it reaches A ?

I assume this means it is OK for the rocket to crash onto the surface of B.
What you haven't said is how the x coordinate is laid out. I assume you intended that

4. x = 0 at the center of A and 6r at the center of B.​

Now you have

acceleration=GM/x^2 -GM/(6r-x)^2

Think about the direction of the acceleration (force) with respect to x = 0.

Finally

i integrated both sides with limits of 0&Vm(launch speed) for dv

I do not get what your integrals look like. That is, what are the integrands on both sides of the equal sign, and what the upper and lower limits on both sides?

Regards,
Buzz

 

[Buzz Bloom]

Hi @Hashiramasenju:

I just realized I made a mistake in my previous post. Sorry about that.

I assume this means it is OK for the rocket to crash onto the surface of B.
What you haven't said is how the x coordinate is laid out. I assume you intended that

4. x = 0 at the center of A and 6r at the center of B.​

This should have been:

I assume this means it is OK for the rocket to crash onto the surface of A.
What you haven't said is how the x coordinate is laid out. I assume you intended that

4. x = 0 at the center of B and 6r at the center of A.​

Regards,
Buzz

 

[Hashiramasenju]

Hi @Hashiramasenju:

I just realized I made a mistake in my previous post. Sorry about that.

I assume this means it is OK for the rocket to crash onto the surface of B.
What you haven't said is how the x coordinate is laid out. I assume you intended that

4. x = 0 at the center of A and 6r at the center of B.​

This should have been:

I assume this means it is OK for the rocket to crash onto the surface of A.
What you haven't said is how the x coordinate is laid out. I assume you intended that

4. x = 0 at the center of B and 6r at the center of A.​

Regards,
Buzz