How far can a golfer hit a drive on Mars… and the Moon?

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This week, undergrad Noah Stanton takes on a burning question in comparative planetology: how would a change of venue to another planet or Moon affect one's golf game? Read on for his deep-dive! Reminds me a little of some of the tangents I followed in my earlier years!

by Noah Stanton

Have you ever been watching golfers playing on Pebble Beach and thought, ‘What would happen if he or she took that shot on Mars or the Moon?’. I’d assume no, but I am here to tell you this important information. In order to figure out a way to model a golfer’s shot let’s start somewhere we know a little bit better, Earth. 

Modelling a golf shot involves bio-mechanics, aerodynamics, elasticity of the golf ball, etc., which is maybe outside the abilities of a mere blogpost. I will need to make some assumptions, focus on some parts of the swing, and ignore the rest. The two modeled parts of the swing will be:

1)    The initial contact and acceleration due to the club-head hitting the ball
2)    The flight of the ball after the initial contact

The coordinate system for the shot is oriented like this:

With the y-axis representing the height of the ball.

Initial Contact

Normally when a golfer’s golf club-head hits the golf ball, the golf ball compresses significantly. (link to high-speed shot of golf ball hitting steel plate: https://www.youtube.com/watch?v=00I2uXDxbaE) We will have to assume that this collision is completely elastic. All of the energy the golfer puts into his golf swing is transferred into the ball.

The initial acceleration of the ball to a simplified assumption of:

Where the final velocity was the speed of the ball just after it was not in contact with the club-head and the initial velocity was simply 0 m/s in both the x and y direction.

dt for this model was just the duration of time the ball was in contact with the club-head. Club-head contact duration was not the easier parameter to find, but Roberts et al. (2019) luckily had already measured this parameter for me. Roberts et al. (2019) puts the contact duration at about 420* 10^-6 seconds for an elite golfer’s club-head speed of 180 mph (debatable many sources tell otherwise). This initial shot was also assumed to only have velocity in the x and y direction with no side velocity in the z direction.

Flight of the ball after the initial contact

This was the hardest part of the whole model. When a golf ball model trajectory includes the spin of the ball and air resistance a number of things come into play. To show this, let’s look at the set of equations I used to model the ball from J.W.M. Bush at MIT’s ‘The aerodynamics of the beautiful game’.

Simplifying this equation to my assumptions of only backspin (spinning only around the z-axis) and no z direction velocity, or acceleration, and dividing out the mass, m, we get:

And

With r and m being the radius and the mass of the golf ball respectively. ρ is the density of the air
V_x and V_y are the x and y velocities, and ω_z is the angular velocity in the z direction. It should be noted how the second term in the a_x equation is negative. This is due to the fact that cross product of the angular velocity and the linear velocities (in x,y,z directions) is -ω_z*V_y with the above assumptions.

The reason why the spin or angular velocity of the ball is important is an effect known as the magnus effect. To quote J.W.M. Bush, “The Magnus Effect is the tendency of a spinning, translating ball to be deflected laterally, that is, in a direction perpendicular to both its spin axis and its direction of motion.” The z-axis negative spin deflects laterally in the positive x-direction. A good visualization of this phenomena can be seen in this video (https://www.youtube.com/watch?v=2OSrvzNW9FE).    

C_D and C_L are the drag and lift coefficients, respectively, of the golf ball. These coefficients are determined empirically meaning that they can only be determined experimentally. Again, luckily Bin Lyu et al., (2018) have gone ahead and made those measurements.
    

Without getting into too much detail, Bin Lyu et al., (2018) came up with best-fit quadratic approximations for the two coefficients. 

Drag Coefficient

“High-speed”

"Low-Speed"

With Re being the “Reynolds number” – a parameter that helps predict the flow of air around the ball (like whether it will be turbulent or smooth (laminar)). The high-speed/low-speed boundary is at around 80,000 Reynold’s number.

Also Re=  VD/ν with V being linear velocity, D being the diameter of the golf ball, and v being the kinematic viscosity of air.

Lift Coefficient

With S being the “spin parameter” defined as:

with ω being the angular velocity, r being the radius, and V being the linear velocity. Using the Reynold’s number and the spin parameter I can have the two coefficients change over time due to the decaying spin of the ball and the changing velocity of the ball over time. 

Initial values for the model

I am almost ready to run this model, but what should I use to initialize the earth, mars, and moon shots? 

These values are pulled from PGA golf ball regulations, and from a Golfworld article.

*** Mars absolute viscosity was approximated by summing the absolute viscosities of all Mars chemical species (approximated to Mars temperatures at Gale crater). This is not an accurate depiction of the absolute viscosity of the Martian atmosphere as firstly this summing method is the least accurate way to determine this parameter. Secondly, though absolute viscosity is barely changed by pressure, the pressure difference from Earth’s atmosphere to the Martian atmosphere is significant. There are measured values of this quantity, but that would be no fun.
**** I could not find any literature on the absolute viscosity of the moon’s exosphere. There is a sound description of its composition, but at the number densities present… I cannot accurately predict this value without a lot more work… too much for this blogpost. I set it to a value somewhere between the Earth and Mars values.

THE FINAL RESULT

The model was close-ish (for lack of a better term) to an elite golfer’s optimal drive carry of around 315 yards on Earth, but did fall 30 yards short. There are a number of reasons why this is possible. First, there could be an issue with the initial input data for the model on which many sources conflicted. With  a more rigorous study you’d probably want to sample your preferred golfer yourself. The second reason for the discrepancy could be related to how I actually calculated the position, velocity, and acceleration in my numerical model. The particular numerical method I employed a simple backward Euler method which is prone to damping (decreasing the wanted output of) this function I am trying to model.

I am not confident with the max distance for the Mars and Moon runs of the model. Though the magnus effect really does increase the max distance of a spinning ball, the lift coefficient is dependent on the absolute viscosity of the atmosphere. Without proper measurements or calculations of these parameters on Mars and the Moon it is hard to say if the lift coefficient is being properly parameterized. As well, the Reynolds number and spin parameter values, relating to the lift and drag coefficients, were empirically derived in Earth’s atmosphere. The quadratic approximations for these coefficients may not be applicable at all in the Martian atmosphere or the Moon’s exosphere.
  

It is important to see why our depictions of physical processes can only go so far. Without the proper input values into my model I can’t be so confident with the output. The model is only as good as the input data and its relevance to the environment I want to model. As well, the method chosen to numerically calculate the physical model will directly change the output. Care needs to be taken when deciding what numerical method to use for different processes.

So after all of this the only advice to golfers is: Golf on the Moon and you’ll significantly improve your long game.

References:

Bush, J W. “Math MIT.” Math MIT (blog). MIT, n.d. http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/11/Beautiful-Game-2013.pdf.

Lyu, Bin, Jeff Kensrud, Lloyd Smith, and Taylor Tosaya. “Aerodynamics of Golf Balls in Still Air.” Proceedings 2, no. 6 (February 23, 2018): 238. https://doi.org/10.3390/proceedings2060238.

Roberts, Jonathan R., Roy Jones, and Steve Rothberg. 2019. “Measurement of Contact Time in Short Duration Sports Ball Impacts: An Experimental Method and Correlation with the Perceptions of Elite Golfers”. figshare. https://hdl.handle.net/2134/11470.