How would I answer number 3?

1. A sumo wrestler originally jumps 10 cm above a trampoline but has the same total energy of a small ball that was 200 cm higher than the trampoline. What causes this equivalency?

2. If a 40 kg gymnast and a 400 kg sumo wrestler each dropped from 1 m above the trampoline, find the final position of each athlete. Assume the trampoline is a simple spring obeying Hooke's law with a k value of 12 000 N/m.

3. Real world trampolines lose energy since they are damped springs with much internal friction. How much energy does the sumo wrestler lose on each bounce in this situation

how do you do number 1

To answer number 3, we need to first calculate the initial potential energy of the sumo wrestler on each bounce and then compare it with the final potential energy after the bounce.

1. Calculate the initial potential energy:
The initial potential energy (PE) can be calculated using the equation PE = m * g * h, where m is the mass of the sumo wrestler, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the initial height above the trampoline.

2. Calculate the final potential energy:
The final potential energy can be calculated using the equation PE = 0.5 * k * x^2, where k is the spring constant of the trampoline, and x is the displacement of the trampoline from its equilibrium position after the bounce.

3. Calculate the energy loss:
The energy loss can be calculated by subtracting the final potential energy from the initial potential energy: Energy loss = Initial potential energy - Final potential energy.

Please provide specific values for the mass of the sumo wrestler, the height, and any other necessary parameters to perform the calculations.

To answer the question, "How much energy does the sumo wrestler lose on each bounce in this situation?", you can use the principle of conservation of energy.

First, let's understand the concept of energy loss in a damped system. In a real-world trampoline, energy is lost due to internal friction, air resistance, and other factors. This results in a decrease in the total energy of the system with each bounce.

To quantify the energy loss, you can use the concept of mechanical energy. Mechanical energy is the sum of the kinetic energy (KE) and potential energy (PE) of an object.

In this case, since the sumo wrestler is bouncing on the trampoline, we can assume that the only two forms of energy involved are gravitational potential energy and elastic potential energy. The equation for mechanical energy can be written as follows:

ME = KE + PE,

where ME represents mechanical energy.

Let's assume that the sumo wrestler starts each bounce from rest, so the initial kinetic energy is zero (KE = 0).

The mechanical energy at the highest point during the bounce (when the sumo wrestler is momentarily at rest) is equal to the gravitational potential energy at the initial height above the trampoline plus the elastic potential energy stored in the trampoline.

ME_highest = PE_initial + PE_trampoline.

Now, suppose the sumo wrestler loses E_loss amount of energy on each bounce. The mechanical energy at the highest point after the loss of energy would be:

ME_highest_loss = ME_highest - E_loss.

Since the gravitational potential energy is directly proportional to the height, and the loss of energy happens on each bounce, we can assume that the gravitational potential energy loss is equal to E_loss.

Thus, the sumo wrestler loses E_loss amount of energy on each bounce.

To calculate the exact value of energy loss, you would need specific numerical values, such as the initial height above the trampoline and the properties of the trampoline itself. However, once you have those values, you can substitute them into the equations provided and calculate the energy loss.

I did #1

#2
potential energy lost = m g h
potential energy gained = (1/2) k x^2

for 40 kg

40 * 9.81 * (1+x) = (1/2)(12,000) x^2

solve quadratic for x

#3
I have no idea because you did not tell me ratio of bounces