Heavy Shadows?

Question

This is a somewhat advanced question. Suppose that two elastic balls moving through three-dimensional space collide and rebound, conserving both kinetic energy and momentum. The balls cast shadows, and these shadows, moving over a flat two-dimensional surface come together, collide and rebound. Now pretend the shadows have mass. Suppose the the shadow of a ball has a mass porportional to the mass of the ball. Then, the colliding shadows:

a) conserve kinetic energy
b) conserve momentum
c) conserve both kinetic energy and momentum
d) conserve neither kinetic energy nor momentum

The answer is b. The shadows cannot conserve kinetic energy. Suppose the balls move straight upwards simulataneously (one on the left, the other on the right) with some space spearating them (I hate that I can't draw it for you to illustrate). Then, at a certain height the ball on the left shoots to the right while simultaneoulsy the ball on the right shoots to the left, resulting in both balls colliding. After the collision, the ball that was on the left shhots downwards while the ball that was on the right shoots upwards. Meanwhile, their respective shadows are always directly below them. Then the shadows are stationary after the collison (as one ball shoots upwards whle the other shoots downwards), so enrgy is conserved.

Now for momentum. The momentum of the balls before collision adds up to some total momentum vector P1 and after collison the two momenta add up to P2. By conservation of momementum P1=P2. The shadow or projection of P1 on the flat surface below is p1 and ...

the shadow of P2 is p2. Then p1 = p2. Why? Because the shadows of parrallel sticks (or vectors) of equal length are equal. What if the balls were not perfectly elastic? Would shadow momentum still be conserved? Yes. All collisions -- elastic or ineleastic -- conserve momentum, but only eleastic collisions conserve kinetic energy. So what does it mean if shadows conservve momentum? It means we can think of a momentum like P as made up of components, like a horizontal and vertical component or an X and Y component, and EACH of these components is itself conserved, just as if the other component did not exist.

Answer 1

I'll go for b. momentum.
In 3D, momentum, a vector, is conserved in each axis, so any projection will show conservation in the projected axes. However, energy is a scalar. Two equal and opposite momenta cancel but the energies of the associated bodies add. While the total of the mv products remains constant, in non head-on collisions the velocities become differently distributed among the axes, so the total mv^2 in a given axis does not stay constant. Taking lin's last collision as an example, it's true that both shadows have momentum and energy in the horizontal plane, but the two mv products have equal magnitudes and are in opposite directions, so total momentum in that plane remains 0.