Is the Phase-Space of a Bouncing Ball a Straight Line?

[Sekonda]

Hey,

The phase-space, a graph of momentum against position, shows a trajectory of a particular system and any point on this trajectory gives a microstate of a particular macrostate; given the Energy of the system is constant... I think this is roughly true, correct me where I'm wrong please!

However the question concerns a ball bouncing between two walls placed at positions ±q, the ball interacts elastically with the walls and travels at a constant velocity. Therefore the energy is constant and magnitude of the momentum is conserved.

So I reckon the Phase-Space of such a system would simply be a straight line from -q to +q at a particular momentum p and also the same line at -p (for the ball bouncing back in the opposite direction)

Would this be correct? If not any help would be appreciated!

Cheers,
Tom

 

[Redbelly98]

Yes, your description is correct.

 

[Sekonda]

Woo! I like being correct, Thanks!

 

[Redbelly98]

You're welcome!

If you'd like to take it a step farther, think about how the phase-space diagram is modified for a deformable ball: when it hits the wall, it actually slows to a stop down over a short distance, then rebounds (speeds up over the same short distance) with the same velocity.

 

[Sekonda]

Hmm that may be similar to the next problem on my work sheet which asks to consider inelastic collisions.

Would the phase-space trajectory, over the short distance, rapidly decline to a zero momentum and then rapidly rise to the same momentum but negative (or opposite sign)?

 

[Redbelly98]

Hmm that may be similar to the next problem on my work sheet which asks to consider inelastic collisions.

Would the phase-space trajectory, over the short distance, rapidly decline to a zero momentum and then rapidly rise to the same momentum but negative (or opposite sign)?

Yes. So those two horizontal line segments would be connected.

Inelastic conditions are a different question though.

 

[Sekonda]

Cool, in the inelastic condition it's losing energy and assumed to be over an infinitesimally small time interval - so I think we just get lines from -q to +q which occur over a number of momenta values that are ever decreasing.

 

[Redbelly98]

Yup, you got it.

 

[Sekonda]

Cheers man, thank again!