Collisions

Question 1

How do you find the relative CoM frame velocity in terms of the momentum before and after 2 particles collide (one into the back of another)?

Answer 1

p1 + p2 = q1 + q2, where p and q and the momentum before and after. Then reference all velocities to a coordinate system moving at speed vcm = (p1+p2)/(m1+m2) = (q1+q2)/(m1+m2) = (m1u1 + m2u2)/(m1+m2)

Question 2

What do the velocities become in terms of the CoM frame?

Answer 2

u1' = u1 - vcm
v1' = v1-vcm

Question 3

What does the conservation of momentum equation become?

Answer 3

p1’+p2’ = 0 = q1’+q2’

Question 4

What is the equation for the velocities if the collision is completely inelastic (all KE lost)?

Answer 4

q1’=q2’=0, so v1 = v2 = vcm = (m1u1+m2u2)/(m1+m2)

Question 5

How do you work out the velocities of the collision if its elastic and no KE is lost?

Answer 5

Can write KE as T = p^2/2m. Use this to conserve momentum before and after. Then set p’ and q’ as |p1’|=|p2’| and the same for q.

Find that the velocities reverse in direction but keep the same magnitude in the CoM frame.

Question 6

How do you convert the velocities in the CoM frame back into the lab frame?

Answer 6

Use the equations v1 = v1’+vcm etc and work out vcm using other equation.