Collisions

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Collisions is the ninth and final lecture in the Mechanics section of PH1011. It covers elastic and inelastic collisions in one dimension, and Galileo Transformations.

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Elastic and Inelastic Collisions
Inelastic collisions cause the objects to move off as one mass after impact; the momentum is conserved but the kinetic energy is not (some is lost through deformation, heat, sound ect). A perfectly inelastic collision would lose the maximum possible energy.

Elastic collisions are the opposite; they conserve both kinetic energy and momentum, and move off separately after impact. The final velocities are therefore more difficult to calculate as two equations must be used - these must be derived from p1=p2, and total out at
 * v1' = m1-m2 / m1+m2 v1 + 2m2/m1+m2 v2
 * v2' = 2m1/m1+m2 v1 + m2-m1 / m2+m1 v2

Limiting cases help - where m1 is far larger than m2, v1 does not change in collision whereas v2 rebounds with double the speed of impact (v2 having started from rest). If the two are equal then the impacting mass stops moving whereas the stationary mass gains a velocity equal to the impact velocity. Where a very small mass hits a larger one it rebounds with its initial speed, and the hit mass does not move.

Galileo Transformation
The Galileo transformation refers to taking results and expressing them from the reference frame of the centre of mass. As the centre of mass velocity does not change during or after the collision, in an inelastic case the two masses move off along the centre of mass velocity. rjcom = rj - rcom; differentiation of this gives vjcom = vj - vcom. r and vjcom are the positions and velocities on a given object j from the reference of the centre of mass. For an inelastic system v1'com = v2'com = 0 and v2com = -m1/m2 v1com : v1com = m2(v1-v2)/m1+m2. Visually, the two masses approach one another and then come to rest at the centre of mass.

For an elastic system vjcom = vj - vcom again; v1com and v2com are equal as before but vcom' = -vcom now. Visually this results in the two masses approaching each other and then recoiling.

Example
The experiment consists of taping a golf ball to a basketball and dropping them both; the recoil of the golf ball hitting the Earth causes it to rebound is opposite to its incoming velocity, which (according to gravitational potential and energy conservation) is = -(√2gh). The basketball then immediately collides with the golf ball, which gives v2' = 2m1/m1+m2 v1 + m2-m1 / m2+m1 v2. From here the expected height of the golf ball can be calculated in an assumed perfect elastic impact.
 * Dropping a large object attached to a small object

Summary
Elastic collisions conserve energy; inelastic collisions do not. This makes inelastic final speeds easy to calculate, but elastic final speeds must use two separate equations - v1' = m1-m2/m1+m2 v1 + 2m2/m2+m1 v2 and v2' = 2m1/m1+m2 v1 + m2-m1/m2+m1 v2. The Galileo transformation involves taking the results as observed from the centre of mass.