Excerpt from Ch. 26 Bad Golf Made Easy - Trouble-shooting Around the Course

Sidehill putts
Golfers are often left with sidehill putts because green designers like slopes on greens in part because they are megalomaniacal sadists who revel in destroying the spirit of golfers. And in part because the slopes facilitate the drainage of water and increase the difficulty of putting. You might have a ball that breaks left or it may break to the right. It may be slightly or drastically uphill or downhill. Gravity is the force which causes the ball to arc downward on a slope trending towards the center of the earth, i.e., the center of the force of gravity. As the formula below clearly demonstrates, where a=acceleration, G=gravitational constant, M=mass, r=radius of the Earth, acceleration due to gravity is roughly 32 feet per second per second.

You must keep all this in mind as you determine the speed and direction of your putt. There are an infinite number of combinations of speed and direction which will result in your ball falling the last few inches into the cup toward the center of the earth on any given 8-foot putt. The fact that there an infinite number of combinations would lead one to believe these putts could never be missed. However, there are an even greater number of infinite combinations of speed and direction which will not result in your ball falling the last few inches toward the center of the earth on any given 8-foot putt.

If say, you aim 12 inches to the right of the hole, and putt the ball with force of x, it may or may not go in, but if you aim 12 inches to the right of the hole and putt the ball with force of x-5, it might just go in. In other words, a putt aimed 11 inches to the right would have to be struck slightly harder thus increasing its velocity reducing the force of gravity’s effect on the ball as it approached the cup. This can be further complicated by the fact that the farther to the right you aim, you may actually be putting uphill for the beginning portion of the downhill putt. You must calculate the first gravitational effect on the putt for its uphill portion of the journey as well. So the formula is something like infinity #1 (the uphill portion of the downhill putt) plus infinity #2 (the downhill portion of the downhill putt) times 32 feet per second per second. In other words: The standard form is (x - h)2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p. If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis, it has an equation of (y - k)2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.
Sometimes a green will have multiple breaks, sloping to the left and then sloping to right, or vice versa, in which case you have an even greater number of infinite combinations of speed and direction. How one infinity can be greater than another infinity? you ask. Well, this is, I am sure, a philosophical and mathematical conundrum and the source of great speculation (see Georg Cantor, German mathematician on the “cardinality of the continuum” or Liebniz; also see Aristotle’s discussion regarding the potential infinite and the actual infinite; and my question, for example, would be: can more angels dance on the head of two pins than on one pin?”). But on the golf course these infinities are not abstract, they are the difference between a par and a bogey, the difference between having a 12-inch tap in par putt or a 5-foot uphill sidehill comeback putt.
When putting, golfers must also decide on whether to increase the speed of putt slightly thus allowing gravity less time for that 32 feet per second per second to influence the curvature of the line of the putt or to decrease the speed thus allowing gravity more time for that 32 feet per second per second to influence said curvature. A common mistake is to pick the “slow line” and hit it the “fast line” intended velocity or to do the opposite, pick the “fast line” and hit the putt the “slow line” velocity. That seldom works. So don’t do that. And keep in mind that friction of the ball rolling on the green will slow the velocity of the ball thus increasing its break as the force of gravity works its magic. The Standard Friction Equation is:
Fr = μrN
Dividing by N, you can get the equation for the coefficient of rolling friction:
μr = Fr/N
where:
• μr is the coefficient of rolling friction for the two surfaces (Greek letter "mu" sub r)
• Fr is the resistive force of rolling friction
• N is the normal or perpendicular force of the wheel on a surface
As you can see, this is not rocket science.
Occasionally, the force of gravity is greater than the friction of the ball on a downhill sidehill putt and you may actually putt the ball completely off the green. This is embarrassing when it happens, as you clearly either forgot or misapplied the standard equation for rolling friction coeffiecient. Duh. In US Open and other golf tournaments, greens often are virtually frictionless due to the secret application of Teflon mist during the overnight hours. A ball sent in motion will continue with no apparent deceleration whatsoever as the force of gravity is greater than the now frictionless putting surface.
More often, a player fails to recognize the gravity of the situation as happened to Ollie Schneiderjans in the Phoenix Waste Management Tournament in Feb. of 2018. Ollie had a 74-foot putt for eagle after driving the green on the par 4 17th hole. He misjudged both speed and direction and hit his putt too hard and rolled it into a pond as the gravitational forces increased when the ball left the putting surface and accelerated 32 ft per second per second minus the slight deceleration due to the bouncing ball’s friction as it contacted the grass of the fringe down the slope leading to the water. After a penalty stroke and a drop, he finished with a bogey. The video of the event shows him pointing to his brain after the putt clearly indicating he had failed to use the above gravitational formula in attempting this putt.
And while it is true that gravitational forces can vary from place to place due to density differences in the crust and mantle of the earth, or the distance from the center of the earth’s gravitational influence, and even on how fast time happens, these differences are minute and likely play no role in making an 8-foot sidehill, downhill, breaking putt.

A tip - avoid the "power lip-out." This occurs when the ball becomes influenced by the micro-gravity well at the time/space continuum event horizon of the cup. On a cup cut into a sloping green, one side of the slope is higher than the other and the difference is a variable [r=radius] in the gravitational acceleration formula above. An eighth of an inch or a quarter of an inch may not seem like much, but when applied to Newton's First Law of Motion in which Force = Mass x Acceleration, it can mean the difference between a birdie and a bogey. As the ball enters the cup it accelerates toward the center of gravity of the Earth and 32 feet per second per second. When (r=3959 miles) you get one result; however, when (r=3959 miles -1/4 inch) you get a different result. Not including that radii differential in your calculations is an egregious error. If a ball is struck ever slightly too firmly and the gravity well event horizon of the cup is unable to overcome the inertial forces of the ball in motion, and you manage to find that one out of an infinite number of combinations of speed and direction, the ball may accelerate as it moves toward the gravitational center of the earth, dip slightly below the planar surface of the opening and if the downward slope of the cup edge is greater than the gravitational forces can overcome, the ball may then escape the gravity well and accelerate away from the hole at a greater speed than which it entered the cup. And instead of making a more or less routine two foot tap-in putt for birdie, you are left with a six-foot comeback putt, albeit a straighter putt for par. NASA often "slingshots" satellites in a similar acceleration maneuver around the sun or planets in our solar system thus applying the gravity well acceleration principles of physics to spacecraft moving among the heavenly bodies. This part of golf may actually BE rocket science as this diagram (Figure 1) clearly illustrates.

In addition, even on level putts, a golfer may perceive the ball lip out of the cup under certain circumstances at a greater speed than that which it entered the cup. This perception may be influenced by the adrenalin pumping into the golfer’s system accompanied by a loud vocalized obscenity as the ball exits the cup. Alcoholic consumption may also be a factor in 1) mis-hitting the putt in the first place, and 2) the above mentioned perception. But this perception is an illusion. Laws of angular momentum (Figure 2)would state that as the lesser diameter areas of the ball rotate against the cup edge, negative acceleration would occur in addition to the gravitational deceleration which would occur as the ball moved away from the center of gravity.

However, laws of angular momentum do apply for a sidehill downhill or uphill putt because when the diameter of the ball’s surface touching the cup edge increases as the ball enters the planar surface of the cup hole, the center of mass of the ball decreases the distance to the imaginary rotational center of the void of the hole, and thus speeds up as if a ballerina were bringing her arms toward her center of mass. In other words, in modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum –) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant as you can plainly see in the accompanying Figure 3.

So with all this in mind, my advice is keep it simple: when faced with that 8 foot sidehill downhill, breaking putt just remember the gravitational acceleration rate, the formula for the arc of a parabola, the potential and actual infinities of speed and direction from which you can choose, the cardinality of the continuum, the law of the conservation of angular momentum, the rolling friction coefficient of the ball on the grass due to grass cut length, the rate at which the grass has grown since last mown, the direction of the grain, the moisture content of the grass blade, the surface wind, the barometric pressure, the air temperature, whether you are playing a 2, 3, 4, or 5 layer ball, the shape and configuration of the dimples of the ball and how that relates to the force you use to contact the ball with the putter surface, the composition of the ball cover and its shock absorbing characteristics when struck with the putter surface, milled, or un-milled; try to avoid striking the ball on the edge of a dimple, avoid the power lip-out, stay calm, exhale, close your eyes, and proceed to attempt the putt forthwith as your playing partners and players standing on the tee box behind you have been watching you figure all this out for some time now are getting impatient with your dithering.