Physics and Long Toss Distance: Velo Predictions


I consulted Dr. Alan Nathan on the topic of long toss, and he pointed me to his trajectory calculator. From this, we’ve determined the ranges of velocites required to throw a ball a certain distance. I’m a long tosser, so this was a really interesting write for me. You can see from the chart in the article how hard you might throw (more accurate measures would require knowing spin rate.)

Figured this one would be a good one to share.


Thanks for sharing Dan



That is an interesting and well-written piece attempting to correlate pitcher’s velocity with long-toss distance.

However, I think there are still issues to consider more deeply with regard to using Alan Nathan’s distance-velocity calculator for your purposes.

One of these is: Pitching velocities are almost exclusively measured as “peak velocity”, also called “release velocity”. Because of the decelerating effect of air resistance, a pitched ball that comes out of a pitcher’s hand with a peak velocity of 90 mph will only be traveling about 82 mph by the time it crosses home plate. So, the average velocity for that pitch would be (90 + 82)/2 = 86 mph.

Prof. Nathan’s calculator takes air resistance into account (otherwise it would give completely unrealistic answers for the real world) but, more importantly for the sake of this discussion, it gives an average velocity required for a ball, launched at a given angle, to travel a distance of interest.

Since pitchers’ velocities are measured at release it makes better sense to correlate long-toss release velocities with long-toss distance. As mentioned in your analysis the release angle and the spin rate do matter and so it would be very difficult to make a really accurate correlation that fits all pitchers.

However, on an individual basis it is possible to deduce a pitcher’s release velocity correlation with his long-toss distance. You just need to actually measure his release velocity with a radar gun and correlate that to the distance that he can long-toss.

There is a widely-held belief among coaches that a pitcher long-tossing with an optimal trajectory angle needs to release the ball at 90 mph to throw the ball 300 feet. From measurements I’ve made on individual throwers (on mild, windless days) I think that is a pretty good estimate.


I cannot find anywhere on Dr. Nathan’s site where it says his velocity calculations are averages. Are you assuming that it is an average? If not, point me to where it says that.

I would assume (but don’t know for sure) that he took everything into account when making this calculator, including the deceleration of the ball. Otherwise, it would be a useless tool…


Yeah, I think Dr. Nathan’s calculator takes that into effect. You should run this article by him on Twitter; he’s @pobguy there.

Great methodology!


Dan, I just assumed that Dr. Nathan’s calculator used average velocities rather than peak velocities; however, after looking at his math it seems that’s not the case–in fact, he does use “initial velocity” as a starting point and factors in air resistance and many other parameters in order to attempt realism in the calcs.

However, I’m still puzzled by this set of calculated distance-velocity correlations in your article:

300ft: 77-80mph

325ft: 82-85mph

350ft: 85-90mph

375ft: 90-94mph

400ft: 95-99mph

It’s not my experience that guys with peak velocity in the 77 - 80 mph range can long-toss a baseball 300 feet. It would be very interesting and useful to test the validity of these calculations experimentally, although one of the biggest challenges might be in the accurate determination of a thrower’s launch angle…and that is certainly a very important factor in the distance-velocity correlation.

I still believe the best way to understand an individual pitcher’s functional distance-velocity correlation might be to measure it experimentally with a radar gun using a marked field, such as a football field.