How much harder do you need to hit an uphill putt? How much softer do you need to hit a downhill putt? This page will answer those two questions.

To analyze and uphill and downhill putt, it is easiest to use conservation of energy. Do you remember your junior science or your high school physics? In any system of objects, total energy is conserved. Once a putt is struck, it has kinetic energy. That energy is transformed into thermal energy due to friction as the ball rolls along the grass. On very fast greens, friction is small thus a putt rolls a long ways before all of its kinetic energy is transformed into thermal energy. For a flat putt, the conservation of energy equation would be:

kinetic energy before = thermal energy after

0.5mv*v = Fd

where m is the mass of the ball, v is the initial speed, F is the frictional force and d is the distance the ball rolls (the * symbol represents multiplication). The frictional force, F, is equivalent to:

F = µmg

where µ is the coefficient of friction that depends on the roughness of the surfaces and g is the acceleration of gravity (9.8 m/s2 or
32 ft/s2). Hubbard and Alaways measured µ between golf balls and putting greens in a study called "Mechanical Interaction of the Golf Ball", published in Science and Golf III, proceedings of the World Scientific Congress of Golf (p.429-439). Some of their results are in the table below.

Stimpmeter Distance (ft)	7.5	8.5	9.5	10.5
Rolling Friction µ	0.075	0.066	0.059	0.053

Combining the two above equations yields:

0.5mv*v = µmgd

which simplifies to:

0.5v*v = µgd

and thus the distance a putt travels depends on the inital speed of the ball, µ and g.

d = 0.5v*v/ µg

A stimpmeter gives a golf ball and inital speed of about 6 ft/s. If a ball is rolling up a hill, then there is another energy transformation of kinetic energy into gravitational potential energy. Thus, on an uphill putt, the conservation of energy equation would be:

kinetic energy = thermal energy + grav energy

0.5mv*v = Fd + mgh

0.5mv*v = µmgd + mgh

0.5v*v = µgd + gh

where h is the vertical height gained by the ball. Because some of the original kinetic energy of the ball is transformed into gravitational potential energy, it doesn't roll as far. To compensate, the golfer must give the ball more kinetic energy. On a flat putt, the ball would travel a larger distance. On an uphill putt, the extra distance is translated into height instead. Thus, instead of the extra energy translating into distance, it translates into height. The gain in gravitational potential energy is equal to the extra distance the ball would travel on a flat surface. Thus,

µmgd = mgh

µgd = gh

µd = h

d = h/µ

When putting uphill, the ball must have extra speed comparable to travelling an extra distance on a level surface. That extra distance equals the height gained divided by the coefficient of friction. On a medium speed green (stimpmeter = 9.5), a putt that rises one foot must be hit like trying to knock the ball 17 feet past the hole on a level surface (d = 1 ft/ 0.059). Thus if one has a 20 foot putt up a 1 foot rise hill, the ball must be hit so that it would travel 37 feet on a flat green. A golfer must judge the vertical rise of the putt and know the green's speed.

On a downhill putt, the situation is reversed. One would need to try to hit the ball 17 feet short of the hole for the same scenario as above. One would play a 20 foot putt as a 3 foot putt. That's why on many downhill putts, especially on fast greens, it's difficult to stay short of the hole.

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