A little girl with a mass of 33 kg jumps on a trampoline that has an elastic constant of 2336 N/m. When she jumps onto the trampoline it sinks 81 cm. Assuming that all of the energy of the trampoline launches her into the air, how far will she rise?
To find out how far the little girl will rise after jumping on the trampoline, we can use the principle of conservation of energy. The potential energy gained by the girl when she jumps equals the potential energy lost by the trampoline when it is compressed.
The potential energy gained by the girl can be calculated using the equation:
Potential Energy = mass * gravity * height
Where:
- mass is the mass of the girl (33 kg)
- gravity is the acceleration due to gravity (estimated as 9.8 m/s^2)
- height is the distance the girl will rise
The potential energy lost by the trampoline when it is compressed can be calculated using the equation:
Potential Energy = 0.5 * k * x^2
Where:
- k is the elastic constant of the trampoline (2336 N/m)
- x is the distance the trampoline is compressed (81 cm or 0.81 m)
Since the potential energy gained by the girl is equal to the potential energy lost by the trampoline, we can set up the equation:
mass * gravity * height = 0.5 * k * x^2
Now, we can plug in the known values and solve for the height:
33 kg * 9.8 m/s^2 * height = 0.5 * 2336 N/m * (0.81 m)^2
Simplifying the equation:
323.4 * height = 949.3
Dividing both sides by 323.4:
height = 949.3 / 323.4
Evaluating the result:
height ≈ 2.937 meters
Therefore, the little girl will rise approximately 2.937 meters after jumping on the trampoline.