The Physics of Golf: How Science Can Improve Your Putting
Golf is a very analytical game. There are constant discussions of club head speed, green speed, ball aerodynamics, and other things. The physics of golf is simultaneously simple yet fascinating. We want to show you how you can use the very basic physics of golf to help understand how putting works.
Don’t worry, this won’t be a physics lesson. We’ll do all the physics and math for you and they’ll be hidden in the code we used to generate the graphics in this article. For those interested, that code is hosted on TheDataJocks GitHub in the Golf Physics Repo.
The main goal of this article is to touch on the differences between uphill and downhill putting. One of these scenarios is easier than the other, but it might not be the one that you think! We’re going to apply pure high school physics to study putting in golf.
All of our analysis will be accompanied by helpful graphics to help you see the physics at work. The first section will describe conceptually the forces (literally) at play without going into the mathematics surrounding the physics of golf.
The Physics of Golf: Initial Velocity
Putting is entirely about (1) aim and (2) how hard you hit the ball. We’re going to focus only on the second and look at how putting success depends on how well you can determine how hard to hit the ball. The three following gifs show how the initial velocity controls how good you are at putting.
The following three scenarios show the same put with different initial velocities. One is just right, one is too hard, and one is too soft. The too hard and too soft are only about 6% different from the just right shot. But look at how this small difference can change the outcome!
These gifs were made using the real physics of golf. For an in depth read, check out this article. I will summarize things quickly. The next subsection is not necessary to understanding the golf-related conclusions and observations we’ll make later.
Click here to see more about how we made our GIFs in Matplotlib.
Putting and Golf Physics
There are three forces at play. First, the club head interacting with the ball accelerates the ball. Twice as much clubhead force leads to twice as much ball speed – a result of the impulse-momentum theorem.
The second force is static friction. This one is complicated. Because the ball is not gliding (it is rolling), there is no dynamic friction force. Therefore static friction must be the friction force at play. This may feel weird because we’re used to thinking of static friction as resisting sliding not as slowing objects down.
However, remember the ball is rolling. The static friction force actually imparts an angular acceleration on the ball opposite the direction of travel and independent of the ball speed. The size of this force is dependent on how the state of the green. Wetter greens with longer grass resist rolling more and are called “slow greens”. Shorter grass on top of hard ground barely resists the ball at all and is called a “fast green”. Describing a green as fast or slow is merely talking about the coefficient of static friction of the green surface.
The final force at play doesn’t always play a role. If a putt is slightly uphill or downhill then the gravitational force will change how a ball slows down. Gravity aids the ball’s motion on a downhill putt causing the ball to roll further. Gravity impedes the motion on an uphill putt.
Therefore, the net force on a putt can be realized as the sum of the gravitational force and the static friction force. Uphill putts experience very quick deceleration because the forces add together. Two negatives (both gravity and friction decelerate the ball) make for more deceleration. On downhill putts, gravity counteracts some of the frictional deceleration leading to putts traveling further.
Putting Uphill vs Putting Downhill
The physics of golf can immediately be applied to look at the differences in putting uphill versus downhill. Intuitively, my first though was that putting uphill should be easier because the hill acts as almost a backstop for our putt. This lets the putter be more aggressive.
At least, that is the intuition, is it correct?
There are a few ways to look at this problem. First, let’s look at two examples comparing uphill and downhill putting where the golfer hits the ball 15% harder than the “perfect” velocity. Both are on a fairly steep 4.5 degree slope.
Notice that the downhill shot is made but the uphill wasn’t made. This stands in stark contrast to our previously stated intuition, that uphill putts were easier because you had a backstop.
The reason that the downhill putt was successful is that the putt starts at a much lower starting speed. This means that a 15% initial speed increase results in a slower speed at the hole. That is, a 15% overshoot on a light tap is much less painful than a 15% overshoot on a putt with a big windup.
However, downhill putts do have some inherent risk; our original intuition does hold some weight. The downhill putt, though it can start at a slower speed, runs the risk of a significant overshoot. Take a look at the following scenarios where the putts are now both 0.5 meters per second – about 1 mile per hour – faster than the “perfect speed”. Look at how now the downhill putt leaves a much larger residual.
On the uphill putt, the golf ball rolled roughly 5 feet past the hole. On the downhill putt, the error was twice as large. The golf ball went a full 10 feet past the hole.
So, we can summarize our findings as follows:
- Uphill putts have a larger chance of reaching the hole at too fast of a speed and not going in because of their high speed.
- Downhill putts on the correct line will tend to leave larger residuals because the ball decelerates more slowly.
The second bullet point is also true for a putt that is “off-center”. In fact, the probability that a putt is on the right line but is hit too hard is fairly small. The more likely case is that the green was slightly misread. However, in this case the downhill putts will still leave a larger residual.
All in all, the physics of golf tell us that downhill putts will probably be easier to make but will be riskier in that they can lead to larger misses.
But this doesn’t really tell us explicitly how to get better at putting. Next time you’re on the course in your Sunday Swagger, you’re going to need cold hard data and not just gut feelings to do your best. In the next section, we’ll get more specific.
How Hard to Hit your Putts
In the last section, we talked a lot about “perfect velocity” – the initial velocity which results in the ball just getting to the hole. However, because the ball is allowed to arrive at the hole with some residual speed and still be considered “made”, there is actually a range of initial velocities which result in a made putt. We’ll call these allowable velocities. For a putt to be made, two things need to happen.
- The initial velocity needs to be large enough for the ball to make it to the hole.
- The velocity when the ball is at the hole needs to be less than 1.31 meters per second, about 2.9 miles per hour.
We can actually use this information to determine the range of initial velocities which will lead to a successful putt. Recall the kinematic equation v_f^2 = v_0^2 + 2ax . This equation relates the initial and final velocities to the acceleration and distance traveled.
Let’s only consider flat ground for now on a medium speed green. This means a \approx 0.9 meters per second. For a successful putt, the final speed after traveling the entire distance to the hole needs to be between 0 and 1.31 meters per second.
Plugging these numbers in gives the formula for initial speed as a function of distance x to the hole. The minimum initial speed is when the final speed is zero and is given by v_0 = \sqrt{2\cdot 0.9\cdot x} . The maximum allowable initial speed is v_0 = \sqrt{1.31^2+2\cdot 0.9 \cdot x} .
This type of analysis gives us another way to see the differences between putting up and downhill. The first plot below shows the range of initial velocities which lead to a successful putt on level terrain.
Notice how as the length of the putt increases, the putt gets harder. This is most obvious because the height of the green region decreases as the length of the putt increases. This means that a narrower velocity interval is allowable for a putt to be made
Even worse, because you have to hit the ball harder on a long putt, this shrinking velocity interval is even smaller when measured as a percentage of the perfect velocity. It truly is difficult to sink those long putts.
What about the differences between uphill and downhill putts? The previous section tried to argue that uphill putts were harder. Take a look at the following graphic confirming this fact.
On the left is the allowable initial velocities for a made putt on a 4 degree decline. The right shows the same data for an uphill putt. Notice how much thinner the region is for an uphill putt than for a downhill putt. This means that downhill putts allow for more error while still being made.
Even better, because less initial velocity is required for a downhill putt, the allowable percentage error is significantly larger than for uphill putts. This confirms yet again that downhill putts are easier than uphill ones. The only way in which uphill putts are better is in the average length of the residual.
Conclusions
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