The Cool Physics of a Supersonic Baseball

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.

a baseball with arrows denoting the force of air and the downward force of mass and gravity.
Illustration: Rhett Allain

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.