# Home runs and real world projectile motion

1. Jul 12, 2017

### pixel

When Major League home runs are hit now, data is usually displayed from Statcast showing the exit velocity, the launch angle and the range. If your students plug the first two values into the standard projectile range equation, they will find that the result is significantly larger than the reported range. As a recent example, using exit velocity = 109.1 mi/hr and launch angle = 30°, the standard range equation gives 693 ft while the Statcast result is 429 ft. The discrepancy is explained in this article, which discusses the forces acting on the ball besides gravity: http://aapt.scitation.org/doi/10.1119/1.4976652

This might be an interesting topic for classroom discussion.

2. Jul 12, 2017

### Staff: Mentor

The standard projectile motion equation that you refer to assumes that there are no forces on the ball after it's been hit, other than the force due to gravity. Of course, this is a simplified model that doesn't take into account drag due to air resistance or other forces listed in the article.

3. Jul 12, 2017

### pixel

I believe that's what I said. I'm merely pointing this out as a possibly useful classroom discussion relating to a real-world application that students might see when watching MLB.

4. Jul 13, 2017

### Andy Resnick

Thanks for sharing this- I forwarded the article to a lot of colleagues here. I agree, discussing this in Physics I class is a great learning activity.

5. Jul 14, 2017

### Dr. Courtney

Great topic. Thanks for the link. This is a case where one needs to choose between including ALL the info and forces needed for an accurate calculation and adding only the next order approximation to the original approach. There is also the issue of whether one is making a true testable prediction (a priori) that allows validation of the underlying model, or whether one is assuming the model to be correct and treating some of the parameters as adjustable parameters that are determined from the data. Since the approach of the baseball trajectory paper is to assume the model is right and fit for the lift and drag coefficients with the available data, one cannot use the same data set to test or validate the model. Had one determined the drag and lift coefficients in a laboratory or other experiment, one could use the baseball flight data to test or validate the model. This is an important pedagogical point relating to how science works. If you treat parameters in the model as adjustable fudge factors you may not really be testing the model under discussion.

Another issue in play is that the model assumes that the drag coefficient really is constant both in velocity and in air density and also that all the baseballs are identical. Over the course of its flight, most likely the drag coefficient is varying by 5-15%, being highest when the ball leaves the bat (highest velocity) and lowest when the ball has the lowest velocity (either at the peak of trajectory or right before landing.) Fitting to a constant results in an effective average which may reproduce the peak height and distance accurately enough, but which will systematically underestimate the final velocity. Reconstructing an event where an injury or property damage occurs when a projectile lands requires more accurate knowledge of the drag coefficient at the lower velocities and at the actual air density. Conversely, if the final velocity is available through video data, the drag coefficient late in flight can be estimated much more accurately than assuming it to be constant for the whole flight. Most real applications of calculating trajectories including drag use a realistic model of Cd vs velocity.

One final thing to be aware of with the spreadsheet is that one needs to enter the sea level corrected barometric pressure in the appropriate cell. This is not the pressure one would measure with a barometer at the date and location of the event, but rather the pressure that would be reported and recorded by most weather sources. Most real-world trajectory calculators either communicate this more clearly OR have an option to enter either the sea level corrected barometric pressure OR enter the pressure directly measured (often called the station pressure). In practice, we use a Kestrel weather meter to directly measure pressure, temperature and humidity at the site which allows computing the air density to 0.2% or so. Most other scientists and engineers who measure drag or predict trajectories also use site measurements with a reliable device rather than depending on local weather information.

My approach in teaching 1st year physics was to introduce drag at a level where the main take away was an ability to know how accurate all the calculations neglecting it are likely to be.

I might also note that video approaches to measuring drag are inherently inferior to methods that measure velocity directly at different distances. Video is really measuring position vs. time and the rest of the analysis proceeds by computing first and second derivatives. With typical systems, it is hard to get the uncertainty in the drag coefficient below 5%, and 10% is more common. In contrast, if one can engineer an experimental situation where air resistance truly can be neglected (dropping a lead ball over 1 meter), then one can measure g to much better than 1% with a standard video camera. Doppler radar or optical chronographs can more easily determine drag coefficients to better than 1%.

6. Jul 14, 2017

### pixel

I'm glad if someone found this useful. Dr. Courtney's detailed response makes me wonder how the Statcast system was proven out since in a baseball game the ball lands in the stands before reaching it's expected range. I would assume they initially tested in an open field where they could compared measured ranges to predicted ones, over a variety of launch and atmospheric conditions. Also, the ranges are displayed without any +/- errors. It would be interesting to know how accurate they think the results are.