GOLF
The most important shot in golf is the putt. If it takes 2 or 3 shots to get to the green and
another 3 to sink the ball, then the average player will be very unhappy. An interesting
question is whether the chance of putting the ball is entirely in the hands of the player, or
whether the result represents a certain amount of skill plus a certain amount of luck plus a
certain amount of technology built into the putter. If the ball does not collide with the
centre of mass of the putter (it rarely does) then the ball will tend to come off the putter at
an unexpected angle, as shown in the movie of a superball colliding with a block of
wood.
There are other factors that affect the trajectory of the ball on the way to the hole, such as
the slope of the green, as described by Raymond Penner in the Canadian Journal of
Physics, Vol 80, (2002), page 83-96. It is perhaps surprising that it is easier to sink a
downhill putt than an uphill putt, but the reason is simple. The ball curves towards the
hole when rolling downhill, but it curves away from the hole when rolling uphill. The
consequences of missing a downhill putt are worse.
There is an even more obvious reason why golfers sometimes have trouble sinking a putt.
That is, golf balls are not perfectly spherical. They are covered in dimples, so the surface
is quite uneven. As a result, the ball can bounce off a putter at strange angles. To
investigate this effect, I dropped a golf ball vertically onto a flat, horizontal slab of
concrete to see if it always bounced vertically. Sometimes it did, but sometimes it didn’t.
The top surface of the concrete was much smoother than the ball, so non-vertical bounces
can be attributed to the uneven surface of the ball rather than any imperfections in the
surface itself. A non-vertical bounce would be expected if the surface was not exactly
horizontal, or if the ball was not dropped vertically, or if the incident ball was spinning,
but (a) the surface was levelled with a spirit level, (b) the path of the ball was exactly
parallel to a vertical plumbob line and (c) the incident ball had zero spin. Most of the
time, the ball bounced within one degree of the vertical, but a one degree error is quite a
serious problem.
For example, consider a 10 ft putt. The hole has a radius of 5.4 cm and is 304.8 cm away.
On a perfectly level surface, the ball has to be aimed within 1.0 degree of the line joining
the centre of the ball and the centre of the hole. To sink a 20 foot putt, the required angle
drops to 0.5 degrees.
Movie A shows a ball bouncing about 1 degree to the left, and movie B shows a ball
bouncing about 1 degree to the right. The ball was dropped from a height of about 80 cm.
The ball is slightly fuzzy since the ball moved about 5 mm during the 2ms exposure time,
but it is clear that the ball dropped vertically without spin. After the bounce, the ball
acquired a small amount of spin, consistent with the the fact that a horizontal force acted
on the bottom of the ball pushing it sideways. Such a force can arise if there are more
“hills” on one side of the ball in contact with the surface than on the other side. Each hill
can be envisaged as an object pointing out from the ball along a radius, and each hill will
tend to push both downwards and sideways on the surface.
There is another problem when putting. If the ball arrives at the hole too fast, it can
bounce out of the hole when it strikes the opposite lip. This problem was analysed by
Brian Holmes in the American Journal of Physics, Vol 59 (1991), page 129-136.
Alternatively, the ball can run around the top of the hole and then jump out, as shown in
the attached movie. Holmes analysed this situation too.
The end result is that professional putters sink only about half of 6 ft putts, about 1/4 of
10 ft putts and only about 10% of 20 ft putts.
The physics of golf is described in much more detail in a review by Raymond Penner (in
Reports on Progress in Physics, Vol 66 (2003) pp 131-171, and in a book of this title by
Theodore Jorgensen, published by The American Institute of Physics.