Re: collision

In article <d_t8g.25$25.2115@xxxxxxxxxxxxxxxxx>, mmeron@xxxxxxxxxxxxxxxxxx writes:
In article <Yrp8g.56600$d5.210721@xxxxxxxxxxxxxxx>, Gunnar G <debian@xxxxxxxxx> writes:
If two 2D circular shaped objects (with different velocities and masses)
collides total elastically I find that there are four unknown variables,
the two vellocities and their direction in x- and y-direction.
With conservation of momentum (mass times velocity) I get two equations, and
conservation of energy gives a third equation.
Then there is one degree of freedom left. How can I resolve the situation
and find the velocities after the collision?

You can't with the information provided. To see this, consider the
collision in the CM reference frame of the two objects. What you've
before the collision is both objects approaching the CM, from opposite
sides, with the ratio of the magnitudes of the velocities being
inversely proportional to the ratio of masses (follows from the
defintion of CM). After the collision you've them receding from the
CM (which didn't change) due to conservation of momentum), with the
ratio of the magnitudes of the velocities still being the same. And,
after you add the requirement of elasticity, you find that not only
the ratio but, in fact, the actual magnitudes remain the same. So,
the picture after the collision simply looks like the one before, just
run backwards. Only ... the straight line along which the two objects
move after the collision, doesn't have to be the same as the one
before, since rotation of the picture doesn't change the physics. All
such lines are equally consistent with the conservation laws.

So, you need some additional information about the direction of the
momentum exchange during the collision, in order to fix this last
degree of freedom.

If you've ever played pool, the required information is conveyed by
where on the one ball the other ball strikes. To a first approximation,
the impulse that is exchanged is along the line that goes from the
center of the one ball to the center of the other at the instant of
contact.

For perfectly elastic, perfectly rigid and perfectly slippery
[so that we may ignore angular momentum] balls, this first approximation
is exact.

You control the direction taken by the target ball by controlling
where the cue ball hits it. That's your degree of freedom.
.

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