All+the+math+stuff

IT'S MATH TIME!

So, using the graphing program from which those lovely graphs came on the previous page, we found the acceleration of when Harmony's traveling up, and when she's traveling down.
 * Up acceleration on the ground: 7.068 m/s^2
 * Up acceleration on the trampoline: 9.156 ﻿m/s^2
 * Down acceleration on the ground: -10.31 m/s^2
 * Down acceleration on the trampoline: -5.138 m/s^2

The up accelerations are different due to the fact that the surface that Harmony is pushing off of to launch herself up is different. The trampoline is an elastic surface, so it assists her in launching herself up, therefore having a higher acceleration than on the ground.

Now, due to the laws of physics, the down accelerations should be -9.8 m/s^2, because that is the rate at which all things fall. But these numbers are not -9.8. The down acceleration on the ground is quite close, with only about a 5% error, but what explains the drastic difference in the down acceleration on the trampoline? The behavior of the elastic surface of the trampoline could be to blame.

As for the change in momentum, we calculated that all by ourselves. First off, we found harmony's mass to be 64.4 kg. Then we picked points just before impact with the trampoline and just after, which were (1.57 s, 1.54 m) and (1.60 s, 1.55 m). Respectively, they yielded velocities (v = p/t) of -.133 m/s (negative because the path is down) and .388 m/s. Using the known formula for momentum, p = mv, the mometum before the point of impact was -8.5, and after was 24.98, showing a change in momentum of 33.56.

The next logical step in our journey through the physics of trampolines is to look into the energy of the system. Using the same points as before, we did the following calculations to find the total mechanical energy of the system before and after impact: The laws of conservation of energy were observed, as there was only a 1.1% difference between the two => |971.33 - 981.91|/ [(971.33 + 981.91) / 2] = .0108
 * Before: PEo + KEo = Mechanical Energy Initial
 * mgy + 1/2mv^2 = MEo
 * 64.4(9.8)1.54 + 1/2(64.4)-.133^2 = 971.33 J
 * After: PE + KE = Mechanical Energy Final
 * mgy + 1/2mv^2 = ME
 * 64.4(9.8)1.55 + 1/2(64.4).388^2 = 981.91 J

The velocities at the top and bottom of the jump are zero, because at the maximum and minimum heights, there is a brief moment where the object in question freezes and changes direction. Obviously, the object in question here is Harmony.