What is the compression distance of the trampoline when a 31 kg child jumps to a height of 1 m?
The compression distance of the trampoline when a 31 kg child jumps to a height of 1 m is approximately 0.366 meters.
Understanding Conservation of Mechanical Energy
Conservation of Mechanical Energy: Conservation of mechanical energy is a fundamental principle in physics that states that the total mechanical energy in a system remains constant if there are no external forces doing work on the system. Mechanical energy is the sum of kinetic energy and potential energy in a system.
When a child jumps on a trampoline, the trampoline exerts a spring restoring force on the child, leading to the compression of the trampoline. At the highest point of the bounce, all the initial kinetic energy of the child is converted into potential energy.
To calculate the compression distance of the trampoline, we can use the principle of conservation of mechanical energy. The potential energy stored in the trampoline when compressed can be determined using the formula PE = 0.5 * k * x², where k is the spring constant and x is the compression distance.
At the highest point of the bounce, the potential energy stored in the trampoline is equal to the initial kinetic energy of the child:
PE = m * g * h = 0.5 * k * x²,
Substitute the given values into the equation and solve for x:
0.5 * 4550 N/m * x² = 31 kg * 9.8 m/s² * 1 m,
2275 N/m * x² = 303.8 kg*m²/s²,
x² = (303.8 kg*m²/s²) / (2275 N/m),
x² ≈ 0.1337 m²,
x ≈ √(0.1337 m²),
x ≈ 0.366 m.
Therefore, the compression distance of the trampoline when a 31 kg child jumps to a height of 1 m is approximately 0.366 meters.
To delve deeper into the concept of conservation of mechanical energy, you can refer to additional resources and examples on conservation of energy in physical systems.