Line and Speed of ball across the hole determine Sinks.The ball must drop 1/2 its diameter before the front equator hits the back wall of the cup, or else the ball will bounce up and out of the hole. In order for the ball to have sufficient time to drop at least this far once airborne over the cup, the golfer must combine a path across the hole with the ball's lateral speed across the hole so that the path and speed together give the ball the time needed to drop. 

ACTUAL PLOT OF CAPTURE DATA75% of hole diamter available: 50% of hole diamter available: 25% of hole diamter available: The three practical considerations to balance for deciding optimal ball speed rationally: 1. Speed required to traverse surface irregularities (e.g., footprints, ball marks, spike marks, ant hills, leaves, sticks, etc.)  counsels in favor of aggressive, highspeed delivery at the lip. 2. Slowness that maximizes the width of the hole for capture  counsels in favor of minimal arrival speed. 3. Slowness that minimizes the length of a comeback putt in case of a miss  counsels in favor of minimal arrival speed. The VOTE is 2 to 1 in favor of minimal arrival speed, so Voters #2 and #3 search for unanimity by asking #1 if he REALLY, REALLY needs the maximum speed or can accomplish his task with a reduced speed. #1 sheepishly replies that he can live with 2 revolutions per second at the lip, as today's greens are pretty uniform and true surfaces compared to 30 years ago. So #1 votes 2 rps while #2 and #3 vote for 1 rps, and they compromise by unanimously choosing 2 rps. 
SKID  ROLL  DECAY PATTERN FOR ALL PUTTSTypical velocity patterns on different green speeds  11foot putt. Note that all delivery speeds or terminal velocities at the end of the putts converge and are the same, arriving at the 11foot mark all with zero rps. The "Decay" phase of putts sets in at about the same point in the slowing speed of the ball, depending on the green speed. For a given green, with a set green speed, the decay phase ALWAYS sets in at about the same ball speed as it reaches a certain slowness of roll, and the distance from this point to the end of the roll is ALWAYS about the same (ignoring uphilldownhill effects) and ALWAYS takes about the same amount of time. A 20foot putt has a decay phase onset at the same ball speed near the end of the putt as does a 5foot putt, and both putts have a decay phase that lasts about the same length and time after onset. In the above diagram, the "knee" feature at the right end around 10.5 feet means the "decay" phase sets in around 12 ft/s and lasts at least 1/2 foot for all putts on even fast greens and not less than this on slow greens. A "fast" green's decay phase sets in at about 1ft/s (2.3 rev/s), while a "slow" green's decay phase sets in at about 2 ft/s (4.5 rev/s). A fast green's decay phase lasts roughly 1 to 1.5 seconds, and so the ball rolls 2.3 to 3.5 times and covers 5.28" per roll for a total of between 12" to 18" of "decayed" roll. A slow green stops more abruptly and lasts only about 1 second, but even so since the phase begins at a ball speed of around 4.5 rps, the ball still covers as much as 24" in the "decayed" rollout phase. On slow greens, delivering the ball to the front lip just before or at the time of "decay" onset is usually a pretty good choice, but is not that big of a deal when the green is fast and rolls true. Then there is not much of a pronounced difference between the rolling phase and the decay phase. Golfers with good touch always deliver all putts of whatever length to the edge of the lip with about the same terminal velocity or delivery speed. Because of the reality of the physics of ballhole interaction, a delivery speed of around 2 revolutions per second at the lip is a natural and instinctive speed to maximize ball capture on most green speeds today. 

BASIC PARAMETERSMin time for ball to drop at least 1/2 diameter = 0.065 s Ball diameter = 1.68" * 2.54 cm / 100 = 0.042672 m Ball radius = 0.84 in * 2.54 cm / 100 = 0.021336 m Ball circumference per revolution 2 * pi * rad = 1 rev. = 2 * 3.14 * 0.021336 m = 0.13185648 m/rev Hole width = 4.25", 0.086614 m Hole width / Lateral velocity = Time of drop Time of drop = 0.086614 m / 0.13185648 m/rev * X rev/s Time of drop = 0.656880875327 rev / X rev/s Depth of hole = 4 in liner + 1 in dirt = 5 in *.0254 m/in = 0.127 m or 5.95 ball radii, 2.98 balls Max drop time to bottom = 0.160991 s 
Paths and Length of Air Time over CupTop to bottom: 

Capture 
Capture 
Limit of Capture 
Noncapture 
DeepCapture Velocity for Centercut Putt (14 RPS),
but golfer doesn't usually hit a Centercut Putt

Even a Centercut Putt has an Upper Velocity Limit for
Capture (about 10 RPS)

Considering the range of possible velocities for centercut putts, only the range 25 rps is a smart choice, 1 & 6 rps are okay, and 710 rps are not acceptable. Click Any Drawing to ENLARGE 

Allowing a nice margin of error for line of 50% deviation off the center line with the hole effectively only half as wide as it is with a centercut path, 2 rps stands out as the best choice, with the range 13 rps acceptable but 410 rps not. Click Any Drawing to ENLARGE 

CENTERCUT (Hole Width = 0.086614 m 100%, Hole Depth 0.127 m, g = 9.8 m/s^2, Max. Drop Time = 0.160991 s) 

Velocity at Lip (m/s) 
Drop Time (s)Width = 0.086614 m Max = 0.160991 s 
Drop Depth (m)

Drop Width (m)

X rev/s * 0.13185648 m/rev  Width / Velocity  0.5 * g * (Drop Time)^2  Velocity x Drop Time (not > Max Drop Time) 
1 rev/s = 0.131856 m/s = 5.27 in/s  0.656881 s, 100%+  2.114308 m, 5.0", 99.1 ball radii, 100%+  0.0212 m, 0.83", 24.5% 
2 rev/s = 0.263712 m/s = 10.55 in/s  0.328440 s, 100%+  0.528577 m, 5.0", 24.7 ball radii, 100%+  0.0424 m, 1.67", 48.9% 
3 rev/s = 0.395569 m/s = 15.82 in/s  0.218960 s, 100%+  0.234923 m, 5.0", 11.0 ball radii, 100%+  0.0636 m, 2.50", 73.4% 
4 rev/s = 0.527425 m/s = 21.10 in/s  0.164220 s, 102%  0.132144 m, 5.0", 6.19 ball radii, 100%+  0.0849 m, 3.34", 98.0% 
5 rev/s = 0.659282 m/s = 26.37 in/s  0.131376 s, 81.6%  0.084572 m, 3.32", 3.96 ball radii, 66.5%  0.1061 m, 3.41", 100% 
6 rev/s = 0.791138 m/s = 31.65 in/s  0.109480 s, 68%  0.058730 m, 2.31", 2.75 ball radii, 46.2%  0.1273 m, 3.41", 100% 
7 rev/s = 0.922995 m/s = 36.92 in/s  0.093840 s, 58.3%  0.043149 m, 1.69", 2.02 ball radii, 33.9%  0.1485 m, 3.41", 100% 
8 rev/s = 1.054851 m/s = 42.20 in/s  0.082110 s, 51%  0.033036 m, 1.30", 1.54 ball radii, 26.0%  0.1698 m, 3.41", 100% 
9 rev/s = 1.186708 m/s = 47.47 in/s  0.072986 s, 45.3%  0.026102 m, 1.02", 1.22 ball radii, 20.5%  0.1910 m, 3.41", 100% 
10 rev/s = 1.318564 m/s = 52.75 in/s  0.065688 s, 40.8%  0.02114 m, 0.83", 0.99 ball radii, 16.6%  0.2122 m, 3.41", 100% 
50% HOLE WIDTH PATH (hole lateral width = 0.043307 m 50%, depth 0.127 m, g = 9.8 m/s^2) 

Velocity at Lip (m/s) 
Drop Time (s)

Drop Depth (m)

Drop Width (m or %)

X rev/s * 0.13185648 m/rev  Width / Velocity  0.5 * g * (Drop Time)^2  Velocity x Drop Time (not > Max Drop Time) 
1 rev/s = 0.131856 m/s = 5.27 in/s  0.328441 s, 100%+  0.528580 m , 24.7 ball radii, 5.0", 100%+  0.0212 m, 0.83", 24.5% 
2 rev/s = 0.263712 m/s = 10.55 in/s  0.164220 s, 102%  0.132144 m, 6.19 ball radii, 5.0", 104%+  0.0433 m, 1.70", 100% 
3 rev/s = 0.395569 m/s = 15.82 in/s  0.109480 s, 68%  0.058730 m, 2.75 ball radii, 2.3", 46.2%  0.0433 m, 1.70", 100% 
4 rev/s = 0.527425 m/s = 21.10 in/s  0.082110 s, 51%  0.033036 m, 1.54 ball radii, 1.3", 26.0%  0.0433 m, 1.70", 100% 
5 rev/s = 0.659282 m/s = 26.37 in/s  0.065688 s, 41%  0.02114 m, 0.990 ball radius, 0.83", 16.6%  0.0433 m, 1.70", 100% 
6 rev/s = 0.791138 m/s = 31.65 in/s  0.054740 s, 34%  0.01468 m, 0.688 ball radius, 0.57", 11.5%  0.0433 m, 1.70", 100% 
7 rev/s = 0.922995 m/s = 36.92 in/s  0.046920 s, 29%  0.01078 m, 0.505 ball radius, 0.42", 8.4%  0.0433 m, 1.70", 100% 
8 rev/s = 1.054851 m/s = 42.20 in/s  0.041055 s, 25.5%  0.00825 m, 0.386 ball radius, 0.32", 6.5%  0.0433 m, 1.70", 100% 
9 rev/s = 1.186708 m/s = 47.47 in/s  0.036493 s, 22.7%  0.00652 m, 0.305 ball radius, 0.25", 5.1%  0.0433 m, 1.70", 100% 
10 rev/s = 1.318564 m/s = 52.75 in/s  0.032844 s, 20.4%  0.00528 m, 0.247 ball radius, 0.20", 4.1%  0.0433 m, 1.70", 100% 

