Download Theory of Bending Balls (CBP)

How to Curve a Ball
Conceptual and Mathematical Description of the Phenomenon
You may have come across the principle of curving the trajectory of a ball in the following scenario, the
taking of a free kick. The shooter kicks the ball at an angle between the spot and the centre of the goal. But
the ball does not travel in a straight line; it curves to the left around the line of defenders and may beat the
goalkeeper. How does the shooter achieve this curve? Simply by kicking the ball so it acquires a lot of spin
and the spin in the example shown below is in a counter-clockwise sense.
goal
Force
defender wall
velocity
kick direction
spin
ball
The theory of balls moving through a fluid (air) helps us to understand this phenomenon. A moving ball
which does not spin experiences a drag force directed in the opposite direction to the ball’s velocity, but a
moving and spinning ball experiences an additional lift force at right angles to its velocity. This is sketched
in the diagram, above right. The counter-clockwise spin is producing a force 90 degrees in a counterclockwise sense to the ball’s velocity. This is known as the Magnus Effect.
The expression for the force F is quite simple,
𝐹 = 𝐶𝑚 𝑣𝑓
where v is the size of the ball’s velocity and f is the spin rate in revolutions per second. The coefficient 𝐶𝑚 I
shall refer to as the Magnus coefficient and is influenced by the size of the ball, and the density of the air
among other things. The value of this coefficient can be found by experiments.
Coding the Magnus Effect
While this is straightforward, there is one complication. The Magnus force is at right angles to the velocity
at all times, so as the ball curves and the direction of its velocity changes, then so must the direction of the
Magnus force. Let’s assume the ball is travelling at an angle 𝛼 at a particular time and that the ball is
spinning in a clockwise sense. Then the Magnus force is directed as shown below
x
𝛼
𝛼
𝐹𝑥
𝐹𝑦
F
y
We need to find expressions for the forces in the x and y directions, since our code uses forces in these two
directions to compute the dynamics of the ball. Simple trigonometry provides the answers
𝐹𝑥 = −𝐶𝑚 𝑓𝑣𝑠𝑖𝑛(𝛼)
𝐹𝑦 = 𝐶𝑚 𝑓𝑣𝑐𝑜𝑠(𝛼)
which we could write as the following computer code
magnusX = -magnusCoeff*spinSpeed*speed*sin(alpha);
magnusY = magnusCoeff*spinSpeed*speed*cos(alpha);
Explaining the Magnus Effect
Some authors use the Bernoulli principle in explaining this effect. This principle states that faster moving air
has a lower pressure than slow moving air and they may show you a diagram which looks like the one below
This shows a stationary ball with air moving to the left. This is the same as stationary air will the ball
moving to the right. When the ball rotates it drags air with it and so at the top of the ball this dragged air is in
the opposite direction to the air moving past the ball, so reduces the relative speed of the air. At the bottom
of the ball the dragged air speed is in the same direction to the moving air speed so the relative air speed is
higher. So we have slow (high pressure) air at the top and fast (low pressure) air at the bottom. There is a
force downwards from high to low pressure. Since the ball is moving to the right, it curves downwards as we
expect. Unfortunately this argument is not correct, since it is not borne out by experiment.
When we look at an experimental visualisation of fluid flow past a spinning object, we see something much
more dramatic as shown below.
force
velocity
Here we have a ball moving to the left and spinning in a clockwise sense. In the wake behind the ball, the
airflow towards the bottom is “turbulent”, look at all those eddies, but towards the top it is much smoother.
When the ball throws off all those turbulent eddies downwards, they react by pushing the ball upwards, just
like ejecting gas downwards from a rocket engine pushes the rocket upwards. So in the above diagram there
is an upwards Magnus force, so as the ball is moving forwards, it bends to the right.
Here are a couple of computational fluid dynamic simulations which confirm this experimental result.
velocity
Force
Here colors indicate air speed for the ball which is moving downwards in the figure. The ball on the left is
not spinning and the wake of the air behind the ball is symmetrical. The ball on the right is spinning in an
anti-clockwise sense and the resulting assymetrical wake is making it curve to its left.
[C.B.Price 18-02-13]