Marc is working on the newly rebooted Mythbusters show (I’m not,
but wouldn’t that be cool!) working with the crew as a physics consultant. This particular myth has the hosts jumping off a 12-m building to see if it is possible to survive. However, their insurance company made them put in safety precautions and so the hosts built a cushioning system similar to the one shown below. Being the physics consultant, it is Marc’s job to determine if their rig was safe. Marc built a scale model of their rig (seen below) to do tests with. The m0 = 0.15-kg base he used stretched the spring x0 = 0.04 m. He used a M = 0.2-kg lump of putty as a replacement for a human and dropped it h = 30 cm above the bottom of the base.
To determine if the rig built by the hosts is safe, Marc needs to analyze the system and calculate whether the cushioning system can adequately protect the putty (representing a human) from the impact of jumping off a 12-meter building.
Let's break down the given information and use it to solve the problem step by step:
Given:
- Mass of the base (m0) = 0.15 kg
- Spring stretch due to the base (x0) = 0.04 m
- Mass of the putty (M) = 0.2 kg
- Height from which the putty is dropped (h) = 30 cm (convert to meters: 30 cm / 100 = 0.3 m)
- Height of the building (H) = 12 m
1. Calculate the potential energy of the putty at the top of the building:
Potential Energy (PE) = mass (M) * gravity (g) * height (H)
where gravity (g) ≈ 9.8 m/s^2
PE = 0.2 kg * 9.8 m/s^2 * 12 m
2. Calculate the potential energy at the bottom (maximum compression) of the spring:
PE = 0.5 * k * x^2
where k is the spring constant and x is the maximum compression of the spring.
To find the spring constant (k), we can use the information about the base:
k = force (F) / displacement (x0)
The force (F) can be determined using Hooke's Law:
F = k * x0
Since the base is at equilibrium (not moving), the force on the base due to the spring and the force due to gravity acting on the putty must balance out:
0.15 kg * 9.8 m/s^2 = k * 0.04 m
Now we can find the spring constant (k).
3. Calculate the compression of the spring when the putty hits the cushioning system:
Potential Energy (PE) = 0.5 * k * x^2
Solving for x:
0.5 * k * x^2 = 0.2 kg * 9.8 m/s^2 * 12 m
x^2 = (0.2 kg * 9.8 m/s^2 * 12 m) / (0.5 * k)
x^2 = (2.352 kg * m^2/s^2) / k
4. Substitute the value of x0 into the equation to find x:
0.04 m = √((2.352 kg * m^2/s^2) / k)
Square both sides:
0.0016 m^2 = (2.352 kg * m^2/s^2) / k
5. Solve for k:
k = (2.352 kg * m^2/s^2) / 0.0016 m^2
Calculate the value of k.
6. Substitute the calculated value of k into the equation for x:
x = √((2.352 kg * m^2/s^2) / k)
Calculate the value of x, which represents the compression of the spring.
7. Compare the calculated value of x with the height of the building (H = 12 m). If the calculated compression is smaller than the building height, the cushioning system is safe. If the calculated compression is greater than the building height, the cushioning system is not safe.
Using the calculated values, Marc can determine if the rig built by the hosts is safe for someone to jump off a 12-meter building.