I(mass=80 kg) sit down in the center of a trampoline and the surface sinks down by a distance of 10 cm when I reach equilibrium.
How far above the ground in m must the trampoline be placed so that if I jump onto the trampoline from a height of 3 m above the surface of the trampoline I won't hit the ground?
a=9.8 m/(s^2)
0.881
To find the height above the ground at which the trampoline must be placed, we need to consider the principle of conservation of energy.
When you jump onto the trampoline from a height of 3 meters, you have potential energy (PE) which will be converted into kinetic energy (KE). At the highest point of your jump, all the potential energy is converted into kinetic energy, and at the moment you hit the trampoline, all the kinetic energy is converted back into potential energy.
Since you know your mass is 80 kg and the acceleration due to gravity is 9.8 m/s², you can calculate the potential energy at a height of 3 meters using the formula:
PE = m * g * h
where m is your mass, g is the acceleration due to gravity, and h is the height in meters.
PE = 80 kg * 9.8 m/s² * 3 m
PE = 2352 J
Now, when you reach equilibrium on the trampoline, the potential energy is converted back to potential energy at that height. The equation for potential energy is:
PE = m * g * h
where m is your mass, g is the acceleration due to gravity, and h is the height above the ground in meters.
Since the surface of the trampoline sinks down by a distance of 10 cm (0.10 m) when you reach equilibrium, the distance from the center of the trampoline to the ground is 10 cm less than the height above the ground. Therefore, the height above the ground is 10 cm (0.10 m) plus the distance the surface sinks down.
So, the equation for potential energy becomes:
PE = m * g * (0.10 m + h)
Now we can solve for h:
2352 J = 80 kg * 9.8 m/s² * (0.10 m + h)
Dividing both sides by 80 kg and 9.8 m/s²:
h = (2352 J) / (80 kg * 9.8 m/s²) - 0.10 m
Simplifying the equation:
h ≈ 3.0 m - 0.10 m
h ≈ 2.90 m
Therefore, the trampoline must be placed approximately 2.90 meters above the ground to ensure that you don't hit the ground when jumping onto it from a height of 3 meters.