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Can the wind blow a golf ball into the cup?

Dave Tutelman -- Jan 6, 2009
At request of Tommy Craggs -- ESPN The Magazine

On January 6, 2009, I got a phone call from Tommy Craggs, a fact-checker for ESPN The Magazine. He was following up on a "tall tale from sports", a story about golf.

A gambler and golfer tells of a $30,000 putt he had in Las Vegas. It was a twelve-foot putt, but he topped it so badly he thought it would only roll three feet. With a strong helping wind, it rolled all the way to the hole and in. The green was perfectly level, and yes the putt was that badly struck.

Tommy asked if the story was plausible, particularly the ball being blown all the way to the hole. I responded that it could definitely happen if the wind is strong enough. The only questions are how strong is "strong enough", and whether strong enough would ever happen in Las Vegas while people were playing golf.

The computation of how strong the wind has to be is interesting. Almost everything it requires is stuff a college freshman would have to know to pass Physics 101. (The wind resistance of a sphere is not first-semester physics. Everything else is. A couple of years ago, I tutored my son in freshman physics, so I'm pretty familiar with the current contents of the course.)

Approach:

Here are the steps in our analysis:

We need to know how fast the green is. Green speed is measured in a unit called "stimp". Assume the green stimps at 10 or so. (That's fast but not PGA Tour fast. A lot of resort courses stimp at about 10 -- and this took place at the old Desert Inn course, a classic resort setup.) Compute from that how much rolling friction a golf ball will encounter.
We have to find the wind's force on the golf ball. This will not be an absolute number, because it depends on the wind speed, and we don't know the wind speed yet. But if the wind force exceeds the friction force, then the ball, once set in motion downwind, will continue to roll indefinitely. So we need to find force as a function of wind speed.
Find the amount of wind that will exert enough force on a rolling golf ball to overcome the friction we just computed.

We are not looking for an exact answer, just a ballpark number. If the answer is 20mph then the story is easily plausible, if 100mph then not so. The difference between 20mph and 21mph is not going to change the plausibility. Therefore, we will keep the computation simple and ignore minor effects. We will work in MKS units [meter/kilogram/seconds] to keep the formulas easy, and convert to more usual golfing units where we want a feel for the numbers.

1. Find the rolling friction for a 10 stimp green.

A stimpmeter is a nearly-frictionless slanted ramp; here's a video of how they work, 30" long (0.762m) and slanted at an angle of 20°. You roll a golf ball down the track, imparting an initial velocity to the ball. Then you measure how far the ball rolls. That distance, in feet, is the stimp number, or speed, of the green. So, if the green's stimp is 10, the ball will roll ten feet.

The first step in finding the rolling friction is finding the speed of the ball as it leaves the track and begins rolling on the green. The geometry is a right triangle, with the hypotenuse .762m long at an angle of 20° with the horizontal side. So, if the horizontal side is on the ground, the upper corner is 0.261m above the ground. The calculation is:

Height = .762 * sin(20°) = 0.261

All golf balls are very nearly the same weight, 46g. (Or, for our purposes, 0.046Kg.) Computing weight, which is the force exerted by gravity, we get 0.451 Newtons.

Potential energy is weight (force) times height, or

Energy = .451 * .261 = 0.118 Joule

That potential energy is converted into kinetic energy by rolling down the ramp.

Energy = ½ m v² = 0.118 = ½ * .046 * v²

This equation is easily solved for velocity v, which is 2.26 m/s (or 5 miles per hour).

So we have a ball going 2.26 m/s, and friction slowing it down. How much friction will it take to stop the ball exactly 10 feet (3.05m) from the end of the ramp? (Why ten feet? Because that's what a stimp of 10 means.) We can find the force from the old standby

F = ma

We know the mass of the ball. We have to find the acceleration (actually, the deceleration) due to friction with the grass. There is a well-known formula for velocity (v), distance (x), and acceleration (a):

v_final² = v_init² + 2ax

We know that the initial velocity is 2.26m/s, the final velocity is zero, and the distance is 3.05m.

0 = 2.26² + 2a * 3.05

Solving the equation for a, we find that the acceleration is -0.837 m/s². Why negative? Because the force is friction, which acts opposite to the motion. The force decreases the velocity, so the acceleration is a deceleration.

Now we can solve for the force.

F = ma = .046 * .837 = 0.0385 Newton = .0087 pounds

That is not a lot of force. It's about 1/7 of an ounce, or about 4 grams. I guess that's about what we would expect from a fast green -- not much friction on the ball.

2. Find the wind to provide enough force to exceed rolling friction.

The force of wind on any body is:

F = ½ ρ v² A C_d

where:

ρ = density of the air. Use 1.3 Kg per cubic meter for this number.
v = velocity of the wind.
A = area that the body projects to the wind.
C_d = the "coefficient of drag" which depends on the shape of the body.

The golf ball is a sphere of 43mm diameter. (We ignore the dimples for this calculation.) That gives a circular area of 0.00145 square meters seen by the wind. The coefficient of drag of a sphere is 0.47. So the equation above for wind force becomes:

F = ½ * 1.3 * v² * .00145 * .47

Now we have the wind force exerted on the ball, as a function of wind speed v.

3. What wind provides enough force to overcome rolling friction?

We know from step #1 that the force of the wind has to be more than the force of friction if a ball set in motion will continue to roll across the green indefinitely. And have already calculated that friction force to be .0385 Newtons. So the equation above becomes:

.0385 = ½ * 1.3 * v² * .00145 * .47 = .000443 v²

Solving for wind speed v, we get 9.3 meters per second, or about 21 miles per hour.

Real life

Twenty-one miles per hour is a bit windy to be playing golf -- more than a two-club wind -- but I'm sure all of us have played in more wind than that.

However, that isn't the whole story. Wind speed varies with height; the wind is stronger aloft. Our calculations showed what the wind has to be within an inch or two of the ground. But, when we talk about how strong the wind is, we usually mean anywhere from 5 feet (where we live as humans) to 30 feet (the height where some weather stations report their readings). Using a mathematical approximation, we see that the wind speed at these heights can be up to 1.5 times the wind that is affecting the putt. So the story of the windblown putt may require as much as 32mph wind speed, as felt by the golfer or mentioned in a weather report, in order to give 21mph of wind on the ground where the putt is rolling.

Is the story still possible at 32mph? Well, golf is played under such conditions. I played today in nearly that much wind (New Jersey in January -- brrrr!) The Scots and the Irish play in that much wind all the time. But how likely is it in Las Vegas? A trip to the National Weather Service web site shows us that the monthly peak gust in Las Vegas ranges from 58 to 90mph every month of the year. So a 32mph gust when the putt was struck is not at all unlikely. This point of view was reinforced when I shared an early draft of this article with my friend Bob Doyle, also a golfer and retired engineer. In his reply, he mentioned, "I played a Vegas course when the wind was gusting to 50. My two sons and I gutted it out."

More improbable is that the gust is perfectly aligned with the putt. But the wind has to be aligned somewhere, so this may certainly have happened. Given a 4-inch hole and a 12-foot putt, there is 1 chance in 230 that the wind will be aligned well enough to blow the putt into the hole. In fact, our chances become a little better when we take into account that the wind only has to make a 9-foot putt; according to the story, the golfer's mishit got the ball three feet closer without any aid from the wind. That's 1 chance in 180. Again, not outrageous. Combining all the coincidences, we have an event of low probability that:

There are golfers on the course in weather that produces
A 32mph or higher gust,
Which is aligned in a 1 out of 180 direction,
With $30,000 riding on the result of the putt.

But there is nothing in the physics that says it can't happen. And, if you watch sports (or anything else) long enough, just about anything that can happen will happen.

And what about the bottom line for you?

Did you remember enough physics? Would you have been able to answer Tommy if he had asked you? Did you at least follow the solution as you read it?

If it makes you feel any better, I got the answer wrong the first try. I made a numerical error in the simplest part, computing the cross-sectional area of the golf ball. I didn't notice that until a day later.

Minor effects we ignored in the computation

The stimpmeter track's friction is small, but not zero. We treated it as zero.
Some of that kinetic energy from the stimpmeter is rotational (the ball is rolling). This is much less than the linear motion of the ball, so we ignore it.
There is probably a small amount of energy lost at the transition between the track and the ground. We ignore it.
Rolling friction on grass is not perfectly independent of speed. But, with a green as fast as a 10 stimp, we can consider it the same at any speed.
Dimples on the golf ball change its aerodynamic properties. Most of this change is the lift, next is high-speed drag, and low-speed drag is a distant third place. So we will ignore the dimples for this calculation.
The ball is initially going downwind, which reduces the relative speed of the wind and thus the force from the wind. But, as the ball slows down, the computed wind speed gets ever closer to the actual speed relative to the ball. So the computed wind speed will ideally just let the ball continue to creep, not stop altogether.

Last updated -- Sept 2, 2023