This video is just asking to be a physics problem on my final exam. Here we have someone launched into the air with a slingshot. The fellow then opens a parachute and floats back to Earth. I’m not saying I would do this event, but it’s a great physics problem.

### How Fast, How High?

There are two parts to this. First, the slingshot and the jumper accelerating upward due to an interaction with the bungee cord. Second, the jumper in free fall—still moving upward but slowing due to gravitational force.

To determine how fast and how high the jumper goes, we need data. The simplest data to gather is time values from the video. Looking at one of the launches I find that the human is accelerating upward for 0.56 seconds, then moves upward for another 5.8 seconds until reaching his highest point. If you watch carefully, you will notice that the rider leaves the slingshot just about halfway up the length of the cranes.

Let me start with the second part of the motion. If I assume the air resistance is small, then I know two things other than just the time. I know the final vertical velocity is 0 m/s since this would be at the highest point. I also know that the vertical acceleration is -9.8 m/s^{2} because the gravitational force is the only significant interaction. Using this and the definition of acceleration, I can solve for the velocity as the human leaves the slingshot. Note, this is in one dimension so I will be writing scalar equations for the accelerations and velocities.

Now using my value for the free fall time, I get a launch speed of 56.8 m/s (127 mph). Note that this is about 204 kph—the description for the video suggests a speed of 200 kph. Actually, at this speed air resistance *isn’t* negligible. However, since the jumper is moving upward the speed will rapidly decrease and get into the range where the air resistance doesn’t matter so it’s still a fair estimate.

But how high would this take the jumper? Now that I know the starting and final velocities for the free fall phase, I can calculate the average velocity. From the definition of average velocity I can determine the change in height.

Using the value for the starting velocity and the time, I get an increase in height of 164.6 meters or 540 feet. The video says “over 300 feet” and this value agrees with that. Yes, I will make a second calculation that takes into account the air resistance force—but just hold on for now.

### What About the G-Force?

Determining the acceleration during the slingshot launch is a bit more difficult. Bungee cords might seem like giant springs, but they are not. Even if they were springs the force on the rider decreases as the cords become less stretched. This means that there is a non-constant acceleration. There are two ways to approach this acceleration problem. The first is the simplest—and that is to just calculate the average acceleration.

The average acceleration is defined as:

I know the starting velocity is zero and the velocity at the end of this slingshot is 56.8 m/s (from the previous problem). Using the slingshot time, I get an average acceleration of 101.4 m/s^{2} or 10 G’s. Again, this sort of agrees with the video description of “over 6 G’s”.

The second method to calculate the acceleration makes the assumption that the bungee is actually a spring. However, to calculate the acceleration I would need to find the distance the spring is stretched. That’s not so easy to estimate, so I will just leave this alone.

### Modeling With Air Resistance

I won’t go into too much detail, but let me make a comparison between two jumpers—one with an air resistance force and one without. I will use the typical model for air resistance as a force that’s proportional to the square of the velocity. Also, I will use a drag coefficient for a human that is the same as a skydiver with a terminal velocity of 120 mph (54 m/s).

Here is a simple model showing the two jumpers. Feel free to look at the code, but if you want an explanation try checking out this older post. Oh, click the “play” to run and the “pencil” to see the code.

Notice that the jumper with no air resistance goes about 50 meters higher—so I guess that’s a bit significant. OK, I’ll address this in the homework questions below.

### Homework

As I said, this would make an excellent physics final exam problem. Here are some questions that might or might not be appropriate for a test.

- Here is your basic physics test question: Suppose the bungees are like a spring with a spring constant of 2,000 N/m. If a human has a mass of 70 kg and the slingshot is pulled back 20 meters, how high will the human be flung?
- Use video analysis (like Tracker Video Analysis) to examine the motion of the human after leaving the slingshot. Assuming acceleration is -9.8 m/s
^{2}, find the height of the cranes and the distance the bungees are stretched. Note that the far view of the motion seems to be with a wide angle lens—there is likely some distortion. - With air resistance, it would take the jumper less time to reach the maximum height compared to the non-air resistance jumper. This means that the human would need to be launched with a greater speed in order to have air resistance and have a time of 5.8 seconds to get to the highest point. Using the numerical model above, find the launch speed that gives the correct time with air resistance.
- In my estimation, I determined average acceleration. Estimate the highest instantaneous acceleration during launch.
- Design a slingshot that will launch a human 300 meters. Keep maximum acceleration below 10 g’s. For extra credit, take air resistance into account.

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Source: Wired