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The Physics of Jumping: From Humans to The Hulk
This is a great physics problem. Let’s use the work-energy principle to look at the forces on a jumping human. As the title says, after that I’m going to do the same thing except with The Hulk. Hulk smash.
The best thing about using a jumping human example is that we can see two different ways to use the work-energy principle (adapted from the best intro physics textbook — Matter and Interactions, Chabay and Sherwood)
Work-Energy Principle
The work-energy principle basically says that the work done on an on object is equal to its change in energy. Yes, that’s basically just repeating the title “work-energy”. We need to break this down (and then consider what we mean by “object”).
We can define work as a force (F) applied over a displacement (Δr).
Yes, that’s a dot product since work (W) is a scalar and F and Δr are vectors. Don’t worry — we are just going to deal with simple cases of work. This work changes the energy of system. Yes, the “system” is very important. We have a couple of options for the system. First, there is the point-particle system. This assumes the object is just a point mass. In that case, it can only have kinetic energy (K) which we define as:
With a point particle system (PPS), the work done uses the applied force and the displacement of the center of mass of the object. Don’t worry, this will make sense soon.
The second type of system is a real system. A real system not only has dimensions (is not a point) but it can also have some type of internal energy. For a real system, the work done uses the applied force and how far the point of application moves (which can be different than the displacement of the center of mass).
Jumping Energy (change in internal energy)
We can look at a basic jump by considering three instances. First, the person starts in a crouched position (we can call this position 1). Next, there is the point where the human is extended at the end of the jump but still in contact with the ground (position 2). Finally, the person is at the highest point in the jump (position 3). For each position, I can measure the height of the center of mass (or some other…
