One Last Problem, also related to energy

[Antepolleo]

Ok, this is the last time I'll bug you for a while, I promise. Actually I have two questions here.

I cannot for the life of me figure out why these answers are wrong. Here's the first question:

A child's pogo stick stores energy in a spring with a force constant of 2.40 104 N/m. At position A (xA = -0.140 m), the spring compression is a maximum and the child is momentarily at rest. At position B (xB = 0), the spring is relaxed and the child is moving upward. At position C , the child is again momentarily at rest at the top of the jump.

This problem has 5 parts, and I've managed to successfully obtain the answer to four of those parts. Here's the trouble:

(d) Determine the value of x for which the kinetic energy of the system is a maximum.

Now some information is imparted in an eariler question here:

(a) Calculate the total energy of the child-stick-Earth system if both gravitational and elastic potential energies are zero for x = 0.

I got the answer to that just fine. I noted that all associated potential energies are 0 for x = 0. Since this problem never mentions nonconservative forces, wouldn't it be safe to assume, due to the concept of conservation of energy, that the kinetic energy of the system is at a maximum when x = 0? I tried that and it did not work.

 

[M98Ranger]

It's understandable that you're frustrated with this problem, but let's take a closer look at the information given and see if we can figure out why your answer for part (d) may be incorrect.

First, let's review the concept of conservation of energy. This principle states that energy cannot be created or destroyed, only transferred or converted from one form to another. In this problem, we are dealing with potential and kinetic energy, which are both forms of mechanical energy. So, when the child is at position A, all of the energy in the system is in the form of potential energy (elastic potential energy in the spring and gravitational potential energy due to the child's height above the ground). As the child jumps, this potential energy is converted into kinetic energy, reaching a maximum at position C when the child is at the top of the jump.

Now, let's look at the information given for part (d). We are asked to find the value of x for which the kinetic energy of the system is a maximum. This means that we need to find the position at which the kinetic energy is at its highest point. This may not necessarily be at position A (x = -0.140 m), as you suggested. Remember, the child is moving upward at this point, so the kinetic energy is increasing, but it is not yet at its maximum. The kinetic energy will continue to increase until the child reaches the top of the jump at position C, and then it will start to decrease as the child begins to descend.

So, to answer part (d), we need to find the position at which the child has the highest velocity, since kinetic energy is directly proportional to velocity. This position will be at the top of the jump, when the child is momentarily at rest (velocity = 0) and about to start descending. This is why the correct answer for part (d) is x = 0, not x = -0.140 m.

I hope this explanation helps you understand why your previous approach may not have worked. Keep in mind that when dealing with energy problems, it's important to consider all forms of energy in the system and how they are changing as the system moves. Keep practicing and you'll get the hang of it. Good luck!