A 65.7 kg person jumps from a window to a fire net 21.0 m below, which stretches the net 1.14 m. Assume that the net behaves like a simple spring, and calculate how much it would stretch if the same person were lying in it.
To calculate how much the net would stretch if the person were lying in it, we need to calculate the spring constant of the net and then apply Hooke's law.
1. Calculate the gravitational potential energy:
The gravitational potential energy can be calculated using the formula: GPE = m * g * h, where m is the mass (65.7 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (21.0 m).
GPE = 65.7 kg * 9.8 m/s² * 21.0 m
2. Convert the gravitational potential energy into elastic potential energy:
The elastic potential energy stored in a spring can be calculated using the formula: EPE = 0.5 * k * x², where k is the spring constant and x is the amount by which the spring stretches.
EPE = GPE
0.5 * k * x² = 65.7 kg * 9.8 m/s² * 21.0 m
3. Rearrange the equation to solve for the spring constant, k:
k * x² = (65.7 kg * 9.8 m/s² * 21.0 m) / 0.5
4. Calculate the spring constant, k:
k = (65.7 kg * 9.8 m/s² * 21.0 m) / (0.5 * x²)
5. Calculate the amount the net would stretch if the person were lying in it:
To do this, we need to know the mass of the person when lying in the net. Assuming the person's volume remains the same, the mass will not change. Therefore, the amount the net would stretch if the person were lying in it would be the same as when the person jumps.
Plug in the values for m, g, h, and k into the formula for x, and solve for x:
x = sqrt((65.7 kg * 9.8 m/s² * 21.0 m) / (0.5 * k))
By plugging in the calculated value for k from step 4, you can solve for x and find the amount the net would stretch if the person were lying in it.
To calculate how much the net would stretch if the same person were lying in it, we can use Hooke's law, which states that the displacement (stretch) of a spring is directly proportional to the force applied to it.
First, we need to calculate the force exerted by the person's weight when jumping from the window. The force (F) can be calculated using the formula:
F = m * g
where m is the mass and g is the acceleration due to gravity.
Given that the mass (m) of the person is 65.7 kg and the acceleration due to gravity (g) is approximately 9.8 m/s², we can calculate the force:
F = 65.7 kg * 9.8 m/s² = 643.86 N
Next, we need to calculate the spring constant (k) of the net. The spring constant represents the stiffness of the spring and can be determined using Hooke's law equation:
F = k * x
where F is the force applied to the spring and x is the displacement.
Using the given information that the net stretches 1.14 m under the person's weight, we can substitute the force (F) and displacement (x) into the equation:
643.86 N = k * 1.14 m
Now, we can calculate the spring constant (k):
k = 643.86 N / 1.14 m ≈ 564.53 N/m
Finally, to determine how much the net would stretch if the person were lying in it, we can rearrange Hooke's law equation and solve for the displacement (x):
F = k * x
x = F / k
Substituting the force (F) and the spring constant (k) that we calculated earlier, we get:
x = 643.86 N / 564.53 N/m ≈ 1.14 m
Therefore, the net would stretch approximately 1.14 m if the same person were lying in it.