Could you build an air-powered canon to launch a baseball faster than the speed of sound? Apparently, the answer is “yup.” Check out this build from Smarter Every Day on YouTube. It’s pretty awesome. It’s basically a type of air canon. The way these devices work is, you put a ball in a PVC pipe and seal off the two ends with tape. The next step is to pump the air out of the tube (with the ball still in it). After that, you puncture one wall of the tube so that air rushes in and pushes the ball out through the tape at the other end—sometimes at extreme speeds. So, this is mostly like that—but with a baseball.

Incredibly, they got the baseball to go supersonic on the first shot. In order to measure the ball’s velocity (to confirm its speed), they recorded a slow motion video of the ball passing over two sticks a distance of 12 feet apart. If you can get the time it takes the ball to travel from one stick to the next, you can find the average velocity as:

There are some important things that you might accidentally pass over in this equation.

- This is the AVERAGE velocity. If you have a large time interval, it’s possible that the ball’s velocity at the beginning and end of the time interval are different.
- Since this uses a change in horizontal position (Δx), you only get the average horizontal velocity, not the instantaneous velocity.
- Be careful. You might get the right answer if you use the very common “distance over time” formula, but it’s not a very good habit to calculate the velocity this way. It’s the CHANGE in position divided by a CHANGE in time.

In the YouTube clip, they use the slo-mo video to get a ball speed of 1,538.46 feet per second (469 m/s). This is significantly faster than the speed of sound in air—approximately 343 m/s (but this value changes a little bit depending on the air temperature).

Overall, it’s a great view of the supersonic ball. But you know what this means, right? It’s a perfect opportunity for some video analysis. Yup. The basic idea in video analysis is to look at the location of an object in each frame of the video. With an appropriate scale (like the distance between two sticks) you can get the position (x and y) of the object in each frame. If you know the frame rate (we do) then also get data about the time from video. It’s awesome. Oh, I use the free software Tracker Video Analysis. Also, a big hat tip to Destin (from Smarter Every Day) for including a distance and the frame rate right in the video.

So, instead of looking at just the time it takes the ball to travel between the two sticks, I can get the position in every frame. This will be a different way to find the velocity of the ball—a better way such that I can take into account the change in velocity as the ball travels out of the launcher. Once I go through and mark the location of the ball, I get the following plot of the horizontal position as a function of time.

Since this is a plot of position vs. time, the slope of the data will be the change in position divided by the change in time (from the definition of slope). Oh wait. That’s the same as the definition of velocity. So, the slope IS the velocity. See how nice that is? It shows the ball is not moving at a constant velocity. Even better, since I marked the location of the ball in a whole bunch of video frames, I can calculate the slope (the velocity) at both the beginning and end of the motion. Notice that the ball’s speed decreases? That’s because of the air drag. It’s cool that you can actually measure how much the ball slows down. Well, that’s the great thing about a super high speed camera.

If I select a portion of the data at the beginning of the video, I can use a linear fit to determine the slope of the position vs. time which gives the velocity. From this, I get an initial velocity of 456 m/s at a time of around 0.002 seconds. Near the end of the video, the graph has a slope of 382 m/s at a time of about 0.011 seconds. From this change in velocity over this time interval, I can calculate the horizontal acceleration of the ball.

But why does the ball slow down? After the baseball leaves the launcher, there are just two interactions that cause it to change its velocity. There is the downward pulling gravitational force and the backwards pushing air drag force due to the collision between the ball and the molecules in the air.

The gravitational force is usually fairly significant—however, in this case we are looking at a super short time interval such that it doesn’t really cause a large change in velocity of the ball. But what about the air drag? We can build a model for this air drag force that depends on the speed of the ball (v), the density of air (ρ), the cross sectional area of the ball (A) and a drag coefficient that depends on the shape (C). Most of these values are known, but the drag coefficient at high speeds can sometimes be difficult to determine.

OK, I like to say that you don’t really understand something until you can build a model of it—so let’s do that. Of course the motion of this supersonic baseball isn’t so trivial. The air drag force makes the ball slow down—but the air drag force changes with the velocity of ball. But this force decreases as the speed decreases—but that makes the ball slow down less. This means that there is no analytical solution for the position of this ball as a function of time. Our only hope is to build a numerical model.

The key idea of a numerical model is to start with some initial values for the position and velocity of the ball. With the velocity, I can then calculate the force on that ball at that instant. The next trick is to just find the velocity and position of the ball after some very, very short time interval. During this interval, we can assume that the air drag force is constant—it’s at least approximately constant. Then at the end of the short time step, we can use the new velocity to calculate the new air drag force and repeat the whole thing again. Really, the only problem with this method is that instead of one very complicated mathematical problem you get thousands of simpler problems.

No one really wants to do thousands of calculations to determine the trajectory of a ball—so we can just make a computer do the work (they don’t normally complain). OK, I’m going to skip all the programing details and just give you the code. This is both the numerical calculation showing the trajectory of the ball. The red line is the actual data from the video, the blue line uses a drag coefficient of C = 2.5, and the green line is for C = 1.

If you want to look at and change the calculation, just click the “pencil” icon above. Yes, you should play with the program.

But what does this output tell us? It says that if I have a baseball drag coefficient of 2.5, then the model pretty much agrees with the actual data—which is pretty cool if you think about it. Of course, this is only during a very short time interval. As the air drag force continues to slow the ball down, the ball will move below the speed of sound and the drag coefficient will also change. If you the baseball was moving at normal baseball speeds, it would have a drag coefficient around C =0.3. Yes, the supersonic and subsonic air drag forces can be quite different—this is just one of the things that makes fluid dynamics so complicated.

Once you have a model (which I do), you can do stuff with it. Now I can rerun my program for a time period longer than 0.01 seconds and find out what would happen if there wasn’t a wall for the ball to collide with. Of course, I do need to make some type of assumption about the drag coefficient. For this calculation, I am going to let the coefficient be 2.5 while the ball is faster than the speed of sound, and 0.3 when it is slower. That might not be exact, but it will still give us an idea about the motion of this ball.

Also, I am going to start with the baseball launched exactly horizontally with a height of 1 meter above the ground. Here is a plot of the trajectory in that case.

Here you can see that the ball would travel about 5 kilometers before hitting the ground. Also, I should point out that this plot is not a “picture” of the ball’s path since the vertical axis scale is different than the horizontal scale. Still cool.

Now for some homework questions.

- Suppose you fired the baseball straight UP. How high would it go? Do you need to take into account the decrease in air density as you go higher?
- When the ball impacts the target board, it basically gets destroyed. Let’s assume that the final parts of the ball have a zero horizontal velocity (it completely stops). If this ball stops over a distance of 7.6 cm (the diameter of a baseball), what is the average force the ball exerts on the target while stopping?
- How does the momentum and kinetic energy of this supersonic baseball compare to a bullet? You pick the bullet and rifle that it was shot from.
- How far would the baseball travel before moving below the speed of sound?
- In the video, they show a scene of the baseball traveling in front of a Schliren imaging system so that you can see the supersonic shock wave. Measure the angle of this shock wave and use it to estimate the speed of the baseball. Does this give a value near the velocity from the video analysis?
- Build a model that uses a more dynamic value of the drag coefficient. Here, this paper on supersonic meteors might be useful.
- What launch angle would give the greatest range for this baseball (on flat ground)?

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