Now let's figure out what happens when objects collide elastically in higher dimension. Two should be enough for us don't you think? Let's ask what we can learn from eqns. 1.54, which are the equations for energy and momentum conservation . If we're given the initial velocities of the two objects before impact, we'd like to know what they'r velocities are after the collision. That's like we'd like but I'm afraid we're not going to be able to get those completely. What??? How can that be? In one dimension, we saw that applying these equations gave you the final velocities, so what's going wrong?
Well we applied Newton's laws directly, we're certain to get the complete trajectory of the particles, before, during, and after the collision. But that's quite tough. Instead we're using conservation laws. However even though conservation laws are derived from Newton's laws, they aren't equivalent. In general they contain less information so we shouldn't be surprised if they give an incomplete description of what's going on.
So back to the problem at hand. If we want the final velocities of the objects, that means we want to know four quantities since each velocity has both a magnitude and a direction that we wish to know. The problem is that we have only three equations. (It's three equations because the momentum equation is a vector equation, so each component is an equation in its own right. Two components means two equations.) So we have four unknowns and three equations. So we don't have enough equations to find the final velocities.
Does that make sense physically? Sure! If two balls collide with each other, they go off in different directions depending on how they collide. Think about playing pool. Depending on where the moving ball hits the stationary one, it goes off in a different direction. So without more detailed knowledge of the situation, it makes sense that we aren't going to know what the outcome of the collision will be.
Let's consider a particular case, where the masses of the
balls are equal 
. Let's also take
mass 
to be initially stationary, that is 
.
Then cancelling out the m's eqns. 1.54 become
The first equation says the vector sum of the final
velocities is the initial veloicity.
If you represent the two final velocity vectors
and 
as the sides of a triangle,
then 
will be the hypotenuse.
The second equation looks kind of like the Pythagorean theorem.
What kind of triangle obeys this theorem? A right
triangle. So this says that
and are perpendicular!
If you don't follow this reasoning you can get the same result
by taking 
and
squaring it
Subtracting 
gives 
.
So the final velocities are perpendicular.
So you can see the momentum and energy conservation imply an interesting result. Of course we still don't know what final direction the balls will be traveling in. However we know they'll be going in perpendicular directions.
