The Physics of Golf

[Herman Erlichson was a physicist and a historian of science (he had Ph.D.s in both). He was also an avid golfer, and he was my uncle.  He passed away just over a year ago, and I've been wanting to write this column ever since.  We corresponded frequently when I was a teenager, in a time when hand-written letters were still common.  Here is a small anecdote.]

Everyone who plays golf knows that the driver hits the ball the farthest of any club.  It also has the lowest launch angle, or "loft."  Clubs with high loft, such as a sand wedge, pop the ball very high up in the air, but don't hit it very far.

The universe of people who both play golf and also know college-level physics may not be very large, but everyone in this club has puzzled over this conundrum: why is it that a driver has a loft of only about 10 to 12 degrees?  That seems far too low.

Exactly 30 years ago this month, my uncle Herman Erlichson figured this out.  It's the spin.

He published the answer in a serious physics journal [1], but I'm guessing that most golfers don't read physics journals.  So here is what he found.

Everyone in freshman physics learns that the optimal launch angle for a projectile - the angle that makes a ball fly the farthest - is 45 degrees, in a vacuum.  But in the game of golf, 45 degrees is the angle of a pitching wedge, which (as every golfer knows) hits the ball only a short distance, about half as far as a driver.

Now the physics calculation assumes that the ball is in a vacuum, but still: how come the presence of air makes the optimum angle so much lower?  Or as my uncle put it, in his classic understated style:

"The large discrepancy between the approximately 11 deg of loft for the golf driver club and the 45 deg maximum range angle for a vacuum was the motivation to begin a study of the question of maximum projectile range in the presence of air resistance, with particular application to the flight of a golf ball." [1]

The analysis itself is technically very complex, involving 3 forces: gravity, drag (resistance caused by air friction), and lift, caused by the backspin on the ball.  All three are big factors, but the theoretical result of 45 degrees only accounts for gravity.  

Air friction (or drag) turns out to have a quadratic effect, as my uncle showed.  In other words, the drag increases in proportion to the square of the velocity of the ball.  So hitting it harder causes a very rapid increase in drag.  Here's his graph showing how the angle is affected by quadratic drag:

One consequence of "quadratic drag" is that hitting the ball a lot harder only yields a modest increase in distance.  More important, though, is that if we just consider gravity plus drag, the best angle to launch a golf ball is 35 degrees.  Lower than 45, but still nowhere near the angle of a modern driver.  And the distance here is still too low, only 336 feet (112 yards).

My uncle Hymie figured out that backspin makes a huge difference. Backspin generates lift, keeping the ball in the air much, much longer.  My uncle derived equations that allowed him to calculate how the lift force increases with the rate of spin and the speed of the ball.  This produced a very different picture of how far the ball would carry at different angles, shown here:

After accounting for lift, the optimum angle is 16 degrees, and the ball flies about 200 yards.  (This assumes a typical launch speed by the standards of 1983. The much longer drivers used today create a much greater speed off the tee.)  The remaining different between the actual loft of 10-12 degrees can be explained by the fact that for a drive, the teed-up ball is struck just past the bottom of the swing. This makes the launch angle slightly higher than the loft of the club.

There you have it: when you account for all the forces at play, the optimum angle for a golf driver really is around 10-12 degrees.

My uncle Herman Erlichson loved the game of golf and played often, despite having a seriously weakened leg, the after-effect of a polio infection that he contracted in the 1950's.  He might have struggled to master the game itself, but when it came to the physics of golf, he solved a mystery that had puzzled physicist-golfers for decades.

Reference

[1] Maximum projectile range with drag and lift, with particular application to golf

H Erlichson. American Journal of Physics 51:4 (1983), pp. 357-362.

For all 95 of Herman Erlichson's scholarly papers, including his paper on the physics of golf, see his Google Scholar page.