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Sunday, February 25, 2024

The Physics of an Elephant-Powered Slam Dunk

You've been sheltering in place for weeks now, and you could really use some more fun physics puzzles to solve, am I right? Don’t panic. All the physics problems you can imagine are right here on the internet—along with some you never could have imagined. Here's one you won't find in any textbook:

  • A man with a basketball stands on one end of a seesaw. An elephant stomps on the other end. What happens?

Well, naturally, the guy moves through the air and over the elephant—then over another elephant (because where there’s one pachyderm there’s probably another). Then he slam-dunks the ball through a hoop before hitting a bale of hay. You can't make this stuff up. Here's the video—seriously, you have to see this.

Are you thinking what I'm thinking? This is just begging for some physics analysis. Let's get started!

1. Is it real?

Oh, this a great question. As you may have heard, there’s a lot of fake stuff online. To see if this is legit, we want to analyze the motion and see if it follows basic physics models. For that, we need to get some data from the video. Using the Tracker video analysis app, I can find the position of the guy in each frame of the video. Think of x,y coordinates—horizontal movement is in the x direction, vertical in the y direction.

So for a real (not fake) projectile human, there's only one force acting on him after he leaves the flip board, and that is the downward gravitational force (assuming a negligible air-resistance force). In which case, his trajectory should have the following features of standard projectile motion:

  • Constant horizontal velocity
  • Constant vertical acceleration
  • Vertical acceleration of –9.8 m/s2 (near Earth's surface)

Normally, this would be easy to check. I’d plot the horizontal position as a function of time to see if it's a straight line, which would indicate constant velocity. I’d do the same thing for the vertical position to show that it's a parabola associated with constant downward acceleration. But that won’t work here, because the video shifts to slow-motion in the middle, so the time from one frame to the next isn’t constant.

Don't worry. I can fix this. Instead of plotting x vs. t and y vs. t, I will plot y vs. x—this is called the trajectory plot, since it shows the actual position of the guy as he moves through the air. Here’s what I get:

This is the trajectory for the whole thing, including the end where he bounces off the hay bale and flips over. But just looking at the middle of the plot, it does indeed look like a parabola. Tracker also fit a quadratic equation to the data for me, and you can see from the solid curve above that it’s a good fit. Here are the resulting parameters:

Does that mean it's real? Not necessarily. But it’s encouraging; if it didn't have a parabolic shape it would absolutely be fake.

But let's just check some stuff. A general equation for the trajectory of any normal projectile motion looks like this (here’s an old post where I derive this equation):

Note: x0 is the starting horizontal position, which I set as the origin (x0 = 0), so that goes away. θ is the launch angle. Now I can compare this with the fitted quadratic equation from the plot above. First, the coefficient on x, 1.447, equals the tangent of the launch angle, which gives us an angle of 55 degrees. That looks about right.

From the coefficient on the x2 term, –0.1816, I can find the initial velocity v0 (assuming this elephant jump takes place on Earth, with a gravitational field of g = 9.8 N/kg, which seems like a safe bet).

If you don't have a good feel for velocities in meters per second, that's about 20 mph. That's a nice and reasonable speed. OK. I'm pretty satisfied. I think this is a real video. Although this makes me a little worried about this guy. If he thinks jumping over two elephants is a good idea, what other crazy-dangerous stuff will he try?

Now for some more homework questions. I'll give you some hints, but I’m giving you a chance to answer these before I come back to them in the future.

2. What are the acceleration and force during the launch?

Ignoring what happens later, the launch itself could be dangerous too. A giant animal is exerting a significant force on a puny human through the board. Can you estimate the force that the board exerts on the guy during the launch? Also, find the acceleration during the launch phase (so you won't need to know the exact mass of the dude). It will be useful to find the acceleration in units of g's, where 1 g = 9.8 m/s2. So you know, a human can withstand around 20 g's without serious injury. My guess is that this movement is less than 20 g's.

One hint: During the launch motion, he moves just under 2 meters. It's probably better to use distance rather than time when calculating the (average) force and acceleration.

3. What are the acceleration and force on the landing?

This is basically the same question, but on the other end. So how about a twist? What if he landed only on the bale of hay? What would his landing acceleration be if his center of mass moved 60 centimeters during the landing phase? This distance would include the motion of his legs bending and the compression of the hay. Of course, in the video, the hay doesn't fully stop him, but instead just slows him down a little. That's a smart move.

4. Estimate the elephant power.

When the elephant pushes down on the launching board, find the power input. You might want to calculate the force from the elephant pushing down (that’s optional). But even without that, you can still calculate the power. Trust me. Oh, what is the power of an elephant in horsepower, where 1 hp = 746 watts? Yes, the only reason for this question is to set up elephant power as a unit of measure.

5. Correct for the slow-motion segment.

Clearly the middle of the video is in slow motion. This means the playback frame rate is not the same as the real time in between frames. So, what is the playback speed for this video? Let me give a quick example so you understand what's going on here. Suppose the normal playback rate is 30 frames per second. If this was in slow motion at half speed, it would play at 15 fps—but each frame would still represent 1/30 of a second in real time.

So, what is the playback speed in the middle of the video? If you want a bonus challenge, see if you can edit the video to make the motion in real time.

6. How much margin of error was there?

I've already estimated the launch speed at 9.14 m/s at an angle of about 55.35° so that the dude lands on the hay. But it's quite a small target. What if his initial velocity or angle are off by just a little bit? What kind of variation in the initial conditions can this guy have and still hit the target? Is there lots of "wiggle" room, or does he have to be super accurate to avoid a painful crash into the ground?

I don't know. Maybe that’s why the other elephant was there—to save him if he missed.

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