The Joy of Jumping: Calculating Trampoline Stretch Distance
How far does the trampoline stretch when the gymnast is at rest?
Can you determine the distance that the trampoline stretches when the gymnast stands on it at rest?
Answer:
The trampoline stretches 8.22 cm when the gymnast stands on it at rest.
When the gymnast stands on the trampoline at rest, the distance the trampoline stretches can be calculated using Hooke's law. Hooke's law states that the force exerted by a spring is directly proportional to its displacement.
Using the given data:
- m = 52.0 kg (mass of the gymnast)
- g = 9.8 m/s² (acceleration due to gravity)
- h = 2.98 m (maximum height reached by the gymnast)
- d = 0.70 m (distance the trampoline stretches when the gymnast lands)
- k = 6204.73 N/m (spring constant of the trampoline)
- x = distance the trampoline stretches when the gymnast stands on it at rest (to be determined)
According to Hooke's law, the force exerted by the trampoline when the gymnast lands is equal to the force exerted by the trampoline when the gymnast stands on it at rest. Therefore:
mg + k * d = mg + k * x
k * d = k * x
d = x
Therefore, the distance the trampoline stretches when the gymnast stands on it at rest is equal to the distance it stretches when the gymnast lands, which is 0.70 m (70 cm).
Let's bounce with joy knowing that the trampoline is ready for another exhilarating jump!