Re: collision
- From: puppet_sock@xxxxxxxxxxx
- Date: 10 May 2006 11:32:09 -0700
Gunnar G wrote:
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?
What PD tells you is completely correct. He does not tell you how
to do it though.
The trick is to transfer to the centre of momentum frame for the
impact. This is a frame in which the total momentum is zero.
It is zero before the impact, and zero after. And since you've
got total energy conserved, the two masses have to be going
the same speed after the collision, just at different angles.
That is, you've taken care of nearly everything just by transferring
to the COM frame.
So all you need to do is find the relative angle of in and out for
each mass. Then transfer back to the lab frame when you are
done all that.
How do you find the change in angle? Well, that's where what
PD says comes in. When the two particles collide, if you
assume there is no friction going on (so the masses don't
change their rotation, for example) then the change all has
to be impact. And it has to be perpendicular to the surface
of each object.
So consider the first mass, call it A. You find where it contacts
the other mass. Then you draw the line from the centre of
the mass to that point. The component of momentum perpendicular
to that line can't change, as the force is all parallel to that line.
But the total length of the momentum can't change.
We decided that previously. So the component parallel to that
line must just change sign. It's like a reflection.
So, the only hard part is deciding where the disks contact.
That's the impact parameter that PD talked about. Think about
the two masses in COM frame. Their momentums have to be
equal and opposite, and they have to be approaching eachother.
But if they were not on *exactly* the same line, then they
could miss. Projecting their paths forward, you will find that
the paths are some distance apart. If that distance is zero,
they hit head on. If it is more than zero, but less than the
sum of the radii of the two disks, they contact. So, you want
to work out the location of the impact based on the impact
parameter. You then have a keen little problem in geometry.
Socks
.
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- From: Gunnar G
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