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dl447342

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- Homework Statement
- We will now shine a beam of light on a light sail and use conservation of energy and momentum to calculate the force exerted on the sail. We will start with our cube of light (length ##l##, cross sectional area A, energy E, and momentum ##p_* = E_*/c##) moving at the speed of light towards a perfectly reflecting sail (cross sectional area A and rest mass M, initially moving with speed V). We will assume perfect reflection of the cube of light off the light sail (“elastic collision”). After reflecting off the moving sail, why would we expect the magnitude of the momentum of the cube of light to be less than ##p_*##?

a) A perfectly elastic ball bounces back off a very massive truck at rest. Is there any decrease in the magnitude of the momentum of the ball?

b) A perfectly elastic ball bounces off a very massive truck in motion away from the ball. Think about the extreme case that the truck is moving only slightly slower than the ball. Show that there must be some speed of the truck where the ball bounces back with zero velocity.

- Relevant Equations
- Momentum conservation: Let ##V, V', v, v'## denote the initial and final speeds of the truck and the initial and final speeds of the ball, and let ##m, M## denote the masses of the ball and truck where ##m << M##. Then ##mv + MV = - mv' + MV'##.

A) and b) should be useful for solving the initial question.

If the truck is at rest initially, the magnitude of the momentum of the ball becomes ##|mv'|=|MV' - mv|##, but this may or may not be less than the magnitude ##mv##, depending on how large ##V'## is. ##V' = \frac{m(v+v')}{M}## in this case, so it should be about zero. But ##M## is also much larger than ##m##, so I'm not sure what to conclude.

B) If ##V\approx v##, then the final momentum of the ball is ##mv' \approx M(V' - v) - mv##, and using the fact that ##V' = \frac{Mv+m(v+v')}{M}## is just slightly larger than ##v##, I get a similar problem as in part A).

Also, how can I show that there must be an in between speed of the truck where the ball bounces back with zero velocity?

If the truck is at rest initially, the magnitude of the momentum of the ball becomes ##|mv'|=|MV' - mv|##, but this may or may not be less than the magnitude ##mv##, depending on how large ##V'## is. ##V' = \frac{m(v+v')}{M}## in this case, so it should be about zero. But ##M## is also much larger than ##m##, so I'm not sure what to conclude.

B) If ##V\approx v##, then the final momentum of the ball is ##mv' \approx M(V' - v) - mv##, and using the fact that ##V' = \frac{Mv+m(v+v')}{M}## is just slightly larger than ##v##, I get a similar problem as in part A).

Also, how can I show that there must be an in between speed of the truck where the ball bounces back with zero velocity?