Game Development Reference
In-Depth Information
velocity, and spin rate. Further, experiments show that the drag coefficient is also affected
by spin.
For example, consider a golf ball struck perfectly (right!) such that the ball spins about
a horizontal axis perpendicular to its direction of travel while in flight. In this case the
Magnus force will tend to lift the ball higher in the air, increasing its flight time and
range. For a golf ball struck such that its initial velocity is 58 m/s with a take-off angle
of 10 degrees, the increase in range due to Magnus lift is on the order of 59 meters; thus,
it's clear that this effect is significant. In fact, over the long history of the game of golf,
people have attempted to maximize this effect. In the late 1800s, when golf balls were
still made with smooth surfaces, players observed that used balls with roughened sur‐
faces flew even better than smooth balls. This observation prompted manufacturers to
start making balls with rough surfaces so as to maximize the Magnus lift effect. The
dimples that you see on modern golf balls are the result of many decades of experience
and research and are thought to be optimum.
Typically a golf ball takes off from the club with an initial velocity on the order of 76 m/
s, with a backspin on the order of 60 revolutions per second (rps). For these initial
conditions, the corresponding Magnus lift coefficient is within the range of 0.1 to 0.35.
Depending on the spin rate, this lift coefficient can be as high as 0.45, and the lift force
acting on the ball can be as much as 50% of the ball's weight.
If the golf ball is struck with a less-than-perfect stroke (that's more like it), the Magnus
lift force may work against you. For example, if your swing is such that the ball leaves
the club head spinning about an axis that is not horizontal, then the ball's trajectory will
curve, resulting in a slice or a draw. If you top the ball such that the upper surface of the
ball is spinning away from you, then the ball will tend to curve downward much more
rapidly, significantly reducing the range of your shot.
As another example, consider a baseball pitched such that it's spinning with topspin
about a horizontal axis perpendicular to its direction of travel. Here the Magnus force
will tend to cause the ball to curve in a downward direction, making it drop more rapidly
than it otherwise would without spin. If the pitcher spins the ball such that the axis of
rotation is not horizontal, then the ball will curve out of the vertical plane. Another trick
that pitchers use is to give the ball backspin, making it appear (to the batter) to actually
rise. This rising fastball does not actually rise, but because of the Magnus lift force it
falls much less rapidly than it would without spin.
For a typical pitched speed and spin rate of 45 m/s and 30 rps, respectively, the lift force
can be up to 33% of the ball's weight. For a typical curveball, the lift coefficient is within
the range of 0.1 to 0.2, and for flyballs it can be up to 0.4.
These are only two examples; however, you need not look far to find other examples of
the Magnus force in action. Think about the behavior of cricket balls, soccer balls, tennis
balls, or ping-pong balls when they spin in flight. Bullets fired from a gun with a rifling