Collision of Two Circular Objects

Date: 02/02/2002 at 07:13:02
From: Alex Spurling
Subject: Collision of two circular objects

Hi,

I've been trying to solve a little problem I set myself recently. If 
I have two circular objects (coins for example) moving across a 
frictionless surface, knowing their velocity and angles, how can I 
find the angle of each object after they collide? I'm pretty sure 
that the mass of each object does not affect their angles after 
collision, although it does affect the velocity. I have been working 
with velocity vectors to specify the direction and velocity of each 
object. Here's an example: 

C1 is the first object and C2 is the second object. If i is the x 
component, j is the y component, the velocity vector of C1 is 
(2i+1j) and the velocity vector of C2 is (-3i+0j).

However to find the angles in the example above we also need to know 
if the two objects collided head on or if they just skimmed each 
other. This is where I start to get a little confused. This is 
getting hard to explain and I'm not very good at ascii art so I made 
an image:

   

I know this has something to do with the angles at which each object 
is travelling, and the angle theta in the diagram. I can use the 
cordinates of each object to calculate theta, but I'm not sure where 
to go from there.

Thanks,
Alex

Date: 02/02/2002 at 13:06:26
From: Doctor Tom
Subject: Re: Collision of two circular objects

Hi Alex,

It's a good physics problem - not too hard, but not too easy, either.

First, an easy observation: It clearly matters what the masses of the 
objects are. If two coins collide, both of their directions will be 
wildly affected. If a coin and 50-ton cylinder of lead collide, the 
path of the lead will only change microscopically.

For most collision problems, and for this one in particular, it is far 
easier to first convert the system into a center of mass system. If 
you calculate the center of mass of the system at a series of time 
points both before and after the collision, you will find that the 
center of mass moves at a uniform rate in a uniform direction. If V is 
the vector representing the velocity of the center of mass, subtract V
from the velocities of both particles, work out the answers in that 
system, and then add V to the velocities of the answers to get the 
velocities in the real problem.

In a center of mass system, after you have subtracted V from 
everything, your problem looks exactly like this: Two objects are 
moving toward each other in a straight line. They will collide on the 
line, and will continue to move along that line after collision. All 
you need to do is find the velocities after the collision.

To solve a simple problem like this, you need to use two laws of 
physics: the conservation of energy and the conservation of momentum.  
These laws state that the energy of the system before the collision is 
the same as the energy afterwards, and similarly for the momentum.

The energy of a moving object is (mv^2)/2, where m is the mass and v 
is the velocity. Add the two energies before the collision, and the 
result will be the same if you add them after the collision. Energy is 
just a number.

The momentum of a particle is mv, where m is the mass and v is the 
velocity. In general, the momentum is a vector, but since you're in 
the center of mass system, it will just be a signed number in the 
direction of the line in which the two particles are moving. In fact, 
in the center of mass system, the total momentum is zero (that's what 
a center of mass system is, in reality). So if m1 and m2 are the 
masses of the coins and v1 and v2 are the initial velocities, in
the c.o.m. system, m1v1 + m2v2 = 0 (since the objects are travelling 
toward each other, their velocities will be of opposite signs). 
If V1 and V2 are the velocities after the collision, m1V1 + m2V2 = 0 
as well. The m1 and m2 are the same, since the coins didn't change 
mass.

So you need to solve this system of equations:

m1v1 + m2v2 = m1V1 + m2V2 = 0

and

m1v1^2/2 + m2v2^2/2 = m1V1^2/2 + m2V2^2/2.

This is plenty of information, since you know the values of m1, m2, 
v1, and v2. The only unknowns are V1 and V2.

Also, for the coins, it's a slight change in the coordinates since you 
have to keep in mind that the collision will occur not when the 
centers coincide, but when the edges coincide, which will occur when 
the centers are as far apart as the sum of the radii of the coins.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/09/2002 at 05:21:02
From: Alex Spurling
Subject: Collision of two circular objects

Thanks you very much for the reply. I think I understand the relation 
between velocity and mass of the colliding particles and I understand 
how the centre of gravity of the two objects goes in a straight line. 
However, I still have no idea how to calculate the directions that the 
two cylinders take upon collision. I understand that the mass and 
velocity do matter so I can specify Initial Mass, Initial Velocity, 
and Initial Direction (in the form of an angle between 0 and 360 
degrees), but how do I work out how the objects change direction?

Thanks,
Alex

Date: 02/09/2002 at 20:39:40
From: Doctor Tom
Subject: Re: Collision of two circular objects

If one object has mass m and velocity v, and the second has mass M and 
velocity V, then the velocity of the center of mass of this two-body 
system is:

(mv + MV)/(M+m)

Subtract this vector from both v and V, and you will then have a 
system where both masses are coming directly toward each other, toward 
the origin, where the collision will occur. Using these modified 
velocities, use the conservation of momentum and energy from my last 
note, and the objects after collision, will be in the exact same 
directions (or reversed, if the objects bounce) as before the 
collision.

Then, to find the motion in the real world, not in the center of mass 
system, add back the velocity of the center of mass to the two vectors 
you've obtained. These vectors represent the final directions and 
velocities of the objects.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/