Conservation of energy in Gravitation

[phantomvommand]

Suppose a rocket is moving at radial velocity vr and tangential velocity vt in the Sun's gravitational field. At some time, the rocket enters the gravitational field of Mars (with the above mentioned velocities), and gravitation effects due to the Sun can be ignored. After more time, the rocket leaves the g-field of Mars. Let Mars move at velocity vm with respect to the Sun.

The textbook has claimed that in Mars's frame, the energy of the rocket is conserved, while in the Sun's frame, this event is seen as an elastic collision between Mars and the rocket.

I can see how energy of the rocket is solely conserved in Mars's frame, like how objects in Earth's g-field have their energy conserved basically. But how does the "elastic collision" in the Sun frame work? What would be the equations of conservation of momentum/energy? You guys are welcome to introduce new variables to quantify/better illustrate your explanations. Thank you!

 

[A.T.]

But how does the "elastic collision" in the Sun frame work?

A head on elastic collision with an much heavier object means the speed relative to the heavier object is approximately conserved, just the direction is reversed.

Let v and U be the speeds of rocket and planet moving in opposite directions in the sun's frame. If you simplify the trajectory to a U-turn as shown below:

- In the planet's frame the rocket velocity goes from v+U to -(v+U)
- In the sun's frame the rocket velocity goes from v to -(v+2U)

This is a simple Galilean Transformation (subtracting U from both velocities)

From: https://wiki.kerbalspaceprogram.com/wiki/Tutorial:_Gravity_Assist

I can see how energy of the rocket is solely conserved in Mars's frame, like how objects in Earth's g-field have their energy conserved basically.

That is actually an approximation, because Mars is accelerated by the rocket, so its frame is not perfectly inertial. This doesn't matter much for energy, but it does for momentum. You can use the common center of mass frame of Mars and rocket to have conservation of energy and momentum.

But how does the "elastic collision" in the Sun frame work? What would be the equations of conservation of momentum/energy?

Here again you have to use the common center of mass frame of Sun, Mars and rocket, to have conservation of energy and momentum.

 

[Dale]

Note that elastic collisions also conserve energy, by definition.

 

[PeroK]

I can see how energy of the rocket is solely conserved in Mars's frame, like how objects in Earth's g-field have their energy conserved basically. But how does the "elastic collision" in the Sun frame work? What would be the equations of conservation of momentum/energy? You guys are welcome to introduce new variables to quantify/better illustrate your explanations. Thank you!

Another way to describe an elastic collison is one where the separation speed between the two objects is the same before and after the collision. That implies that the KE of each object is conserved in the frame of the other.

 

[A.T.]

Another way to describe an elastic collison is one where the separation speed between the two objects is the same before and after the collision. That implies that the KE of each object is conserved in the frame of the other.

A common analogy is throwing a ball against a wall that is approaching you:
https://solarsystem.nasa.gov/basics/primer/

This is also similar to making an effieicent U-turn with a plane in a moving airmass: