This time a ball is dropped from a height h_0. The partner is the floor, which like the wall does not move.
First the speed on reaching the floor. Gravity is the only thing doing work during the fall, so the mechanical energy is conserved. It is the same equation used in the roller coaster lectures.
Solving \dfrac{1}{2}mv^2 = mgh_0 gives v = \sqrt{2gh_0}. The mass has gone. A heavy ball and a light one dropped from the same height arrive at the same speed.
The figure puts a velocity arrow on the falling ball. It lengthens as the height falls away.
The instant of contact with the floor is exactly the story of the wall. The partner does not move, so the only thing settled is the velocity of the ball.
v' = -ev. The speed is multiplied by e and the direction turns upward.
Momentum is not conserved here. The floor, and the Earth the floor is joined to, receives the difference, just as in the previous lecture.
In the figure the arrow turns around at the floor. The upward arrow is shorter than the downward one, and the ratio between them is e.
After the bounce the motion is under gravity alone again, so the height reached comes from the same conservation of energy.
Put v' = ev into h' = \dfrac{v'^2}{2g}. Since v'^2 = e^2v^2, this is h' = \dfrac{e^2v^2}{2g}.
And \dfrac{v^2}{2g} was the original height h_0. Gathered up, h' = e^2h_0.
The speed is multiplied by e, the height by e^2. The square appears because height is fixed by the square of the speed. At e = 0.8 the speed is eight tenths, but the ball climbs back to only sixty-four hundredths of its height.
The second bounce is the very same story. The ball falls from h_1 and rises to e^2h_1.
So h_2 = e^2h_1 = e^4h_0. It is the same thing repeated, so after n bounces h_n = e^{2n}h_0.
This is a geometric sequence. Each bounce multiplies the height by the same ratio e^2. Whatever height the ball is dropped from, the ratio is unchanged.
e can be measured from this. Since \dfrac{h_1}{h_0} = e^2, measure the height of the bounce and the height of the drop and take the square root of their ratio. A single ruler is all the apparatus required.