A 42.0 kg girl jumps on a trampoline. After stretching to its bottom limit, the trampoline exerts an average upward force on the girl over a displacement of 0.50 m. During the time that the trampoline is pushing her up, she experiences an average acceleration of 65 m/s^2. Her velocity at the moment that she leaves the trampoline us 9.4 m/s[up]. How high does she bounce?
To find the height to which the girl bounces, we can use the principle of conservation of energy. The initial potential energy when she is at the bottom of the trampoline is equal to the kinetic energy when she leaves the trampoline. We can calculate the initial potential energy and equate it to the kinetic energy at the highest point to find the height to which she bounces.
1. Calculate the initial potential energy:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
Since the girl is at the bottom of the trampoline, the height (h) is zero.
Potential energy (PE) = m * g * h = 0
2. Calculate the kinetic energy at the highest point:
Kinetic energy (KE) = (1/2) * mass (m) * velocity (v)^2
Given: mass (m) = 42.0 kg
velocity (v) = 9.4 m/s (upward)
Kinetic energy (KE) = (1/2) * 42.0 kg * (9.4 m/s)^2
3. Equate the initial potential energy to the kinetic energy:
PE = KE
Since the potential energy (PE) is zero, we have:
0 = (1/2) * 42.0 kg * (9.4 m/s)^2
4. Solve for the height (h):
Rearrange the equation to solve for height (h):
h = (KE) / (m * g)
Substitute the values:
h = [(1/2) * 42.0 kg * (9.4 m/s)^2] / (42.0 kg * 9.8 m/s^2)
Simplify:
h = (441.72 m^2/s^2) / (411.6 m^2/s^2)
h = 1.07 m
Therefore, the girl bounces to a height of approximately 1.07 meters.