Physics Problem [RESOLVED]

How do you find the final velocities of two circular objects (think pool balls or pucks on a spark table) on a non-head on a perfectly elastic collision? They do not simply exchange velocities.

I have all masses, velocities, positions, and the angle at which they collide.

Preservation of momentum:
p₁ + p₂ = p'₁ + p'₂ = m₁v₁ + m₂v₂ =
m₁v'₁ + m₂v'₂
Preservation of energy:
E₁ + E₂ = E'₁ + E'₁
.5 (m₁v'₁² + m₂v₂²) = .5 (m₁v'₁² + m₂v'₂²)

You only have to use relative velocities, therefore:
v_1r = 0
v_2r = v₂ - v₁
Then:
v'₁ = v₁ + v'_1r
v'₂ = v₁ + v'_2r

The equations now become:
v_2r =
(m₁v'_1r + m₂v'_2r) / m₂ = (m₁/m₂)v'_1r + v'_2r
and:
v_2r² = (m₁v'_1r² + m₂v'_2r²)/m₂
= (m₁/m₂)v'_1r² + v'_2r²

You can now solve these equations to get v'_1r and v'_2r (too lazy to do it now).

The above story only applies to a head-on collision. If they collide at a different angle, calculate the directional component of the velocites, put that into the equation. Substract the directional component from the total valocity, and add the new (calculated) directional velocity to it.

Are these vectors or scalar values that you are using for your equations?

The velocities are vectors (the masses are of course scalars). By the way, I worked out these equations some time ago, and I found them again:
v'_1r = v'_2r*(2m₂)/(m₁+m₂)
v'_2r = v'_2r*(m₂-m₁)/(m₁+m₂)
and:
v'₁ = (v'₁-v'₂)*(m₂-m₁)/(m₁+m₂) + v₂
v'₂ = (v'₁-v'₂)*(2m₂)/(m₁+m₂) + v₂
these are what you get when you make v₂=0, that's what I did in my old equations, the result should be the same

Ok. I didn't see any notation to indicate that you were using the vector quantities, so thanks for clearing that up. :)

Now how would I find the directional component at time of impact of the velocity.

Is the directional component the vector formed by the centers of the colliding objects?

:confused:

If your velocities are vectors, how could you treat them as scalars (you are multiplying them together.)

Oops, they are indeed scalars, I should think before I write something. :o
Also, the last two equations in my previous post should of course use v1 and v2 instead of v'1 and v'2.

To calculate the directional components you need to do something like this:
s=s₁-s₂ (vector)
s_unit=s/|s| (vector)
v_1,scalar=v₁ . s_unit
same for v₂
now calculate v'_1,scalar and v'_2,scalar
v'₁=v₁ + s_unit(v'_1,scalar-v_1,scalar)
same for v₂

something like this should work, but maybe you need to change something to get the right sign for s_unit

Couldn't you tilt the plane at which you are viewing the collision at some angle once you have the relative velocities so that it becomes a head-on collision?

Also, why didn't you solve for v'1r and v'2r initially (in your first post?)

Did you use the conservation of energy equation? I used the equation for relative velocities:

V2f - V1f = -(V2i - V1i)

How did you solve it using the conservation of energy?

Got a rough going so I am resolving this thread.

Check it out. :)

http://www.vbforums.com/showthread.p...hreadid=256222