How Does a Pogo Stick Utilize Spring Mechanics for Maximum Jump Height?

[CNauert]

General Physics Homework Question! Please help!

A child's pogo stick stores energy in a spring (k= 21000 N/m). At position A (x1=-0.100 m). The spring compression is a maximum and the child is momentarily at rest. At position B (x=0), the spring is relaxed and the child is moving upward. At position C (x2=?), the child is again momentarily at rest at the top of the jump. Assume that the combined mass of the child and pogo stick is 26.0 kg.


a) Calculate the total energy of the system if both potential energies are zero at x=0
Ok, here i got 79.52 J
b) Determine x2.
And here I got 0.312 m
c) Calculate the speed of the child at x=0.
And here I got 2.47 m/s

Here is where I'm having problems!

d) Determine the value of x for which the kinetic energy of the system is a maximum.
e) Obtain the child's maximum upward speed.

 

[Doc Al]

What have you tried so far? Where are you stuck?

Ask yourself: Will the maximum KE, which means maximum speed, occur above or below x = 0? What does energy conservation tell you?

 

[CNauert]

Well since the child is moving upwards, the maximum speed has to hit before x=0 meaning the speed is going to decrease as the child moves upward. So, do I use the state when the child is at x1 = -.100 since that is when the spring will relax and the child's velocity will be at its max??

I'm just confused on where to start with this part of the problem.
I know that the sum of the energy = KE + PEg + PEs. I know that Wnc = Ef-Ei. There are no non-conservative forces though, right? Since friction is not involved? Or is the spring a non-conservative force? If there aren't any, I use Ef=Ei which means KEi + PEgi + PEsi = KEf + PEgf + PEsf.

KE=1/2mv^2
PEg=mgy and
PEs=1/2kx^2

 

[Doc Al]

Well since the child is moving upwards, the maximum speed has to hit before x=0 meaning the speed is going to decrease as the child moves upward.

Good. Since at x = 0 (position B) and above, the only force acting on the child is gravity the speed will only continue to decrease.

So, do I use the state when the child is at x1 = -.100 since that is when the spring will relax and the child's velocity will be at its max??

At x1 = -.100 is when the spring has maximum compression and the KE is zero. So the maximum speed and KE will occur someplace between x = -.100 and x = 0.

I'm just confused on where to start with this part of the problem.
I know that the sum of the energy = KE + PEg + PEs.

Good. Get expressions for each of the PE terms as functions of x.

I know that Wnc = Ef-Ei. There are no non-conservative forces though, right? Since friction is not involved?

Right. All forces are conservative here.

Or is the spring a non-conservative force? If there aren't any, I use Ef=Ei which means KEi + PEgi + PEsi = KEf + PEgf + PEsf.

Good. That can also be written as Total Energy = constant. (Hint: You already figured out the value of the total energy.)

KE=1/2mv^2
PEg=mgy and
PEs=1/2kx^2

Good. Express PEg in terms of x (measured from x = 0).

Use that to express KE in terms of x. Then, how would you find the maximum of some function of x?