- #1

Bladerunner56

- 6

- 1

Hello PF,

I'm looking for physics and math insights that will help me set up and solve a diff. eqn. for the height of a jump from a spring platform. The problem requires a bit of setup explanation, because it appears simple at first, but then, not so simple.

First consider a person standing with knees bent on solid ground. At t=0, the person is at rest. The person then executes a vertical jump by exerting a sinusoidal force to propel the body up. We assume a force amplitude and frequency, and a mass. We'll simplify things by assuming a center of mass (COM) for the person and that the spindly, yet strong legs, have no mass. The position of the COM (initially at a height of 1) begins to move upward immediately and continues to accelerate upward until the force stops or the feet leave the ground. So, at t=0, v, a, and KE = 0, and x=1.

Now consider that the person performs the same jump, but stands on a massless spring platform with spring constant K. Note that this setup is different from most mass-spring problems because there is no initial velocity for the COM, the spring is at equilibrium, and the mass itself is the actuator of the force.

Questions:

1) How does adding the spring platform change x(t) for the COM?

2) Does the value for the spring constant K matter? (assume the spring obeys Hooke's law)

3) There should be an interplay between the leg force, the COM acceleration, and the spring compression (and expansion), but how is this represented in the differential equation?

I tried to solve for x(t) using a second order ODE for a driven mass-spring system, and the result makes no sense. Therefore, I believe I set up the equation incorrectly. Could it be that the leg force is not "external" as in the usual mass-spring setup? If so, how do you work that into the equation?

I'm looking for physics and math insights that will help me set up and solve a diff. eqn. for the height of a jump from a spring platform. The problem requires a bit of setup explanation, because it appears simple at first, but then, not so simple.

First consider a person standing with knees bent on solid ground. At t=0, the person is at rest. The person then executes a vertical jump by exerting a sinusoidal force to propel the body up. We assume a force amplitude and frequency, and a mass. We'll simplify things by assuming a center of mass (COM) for the person and that the spindly, yet strong legs, have no mass. The position of the COM (initially at a height of 1) begins to move upward immediately and continues to accelerate upward until the force stops or the feet leave the ground. So, at t=0, v, a, and KE = 0, and x=1.

Now consider that the person performs the same jump, but stands on a massless spring platform with spring constant K. Note that this setup is different from most mass-spring problems because there is no initial velocity for the COM, the spring is at equilibrium, and the mass itself is the actuator of the force.

Questions:

1) How does adding the spring platform change x(t) for the COM?

2) Does the value for the spring constant K matter? (assume the spring obeys Hooke's law)

3) There should be an interplay between the leg force, the COM acceleration, and the spring compression (and expansion), but how is this represented in the differential equation?

I tried to solve for x(t) using a second order ODE for a driven mass-spring system, and the result makes no sense. Therefore, I believe I set up the equation incorrectly. Could it be that the leg force is not "external" as in the usual mass-spring setup? If so, how do you work that into the equation?

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