Bend it round the wall

The rules of football give the problem its dimensions. For a direct free kick the wall must stand at least 9.15 m (10 yards) from the ball. The goal is 7.32 m wide and 2.44 m high. A wall of players is typically about 1.8–1.9 m tall, and they can jump. So the ball has to clear a high obstacle close by, then still come down under the bar.

The classic plan is to aim outside the wall and let spin bring the ball back in. Three things to balance:

• Speed. Faster kicks leave the defenders less time, but the spin ratio S = rω/v is smaller.
• Spin. More spin means more sideways force, but it also takes more care from the foot.
• Elevation. The ball must be higher than the wall 9.15 m out, yet it must also come back down below 2.44 m by the goal line.

In a simple model of a 28 m/s kick with 8 revolutions per second of side spin from about 25 m, the ball drifts about 2.7 m sideways by the time it reaches the goal line, a very different path from the same kick without spin.

Why does the bend seem to arrive late? Two reasons.

1. The path turns with a radius of 2m / (ρ C_L A), which for a football with C_L ≈ 0.13 is roughly 150 m. A circle bends away from a straight line with the square of the distance, so most of the visible sideways shift happens near the end of the flight: after 80 % of the distance the ball has made only about 64 % of the total sideways shift.
2. The ball slows down, so S = rω/v climbs (the spin loses very little over one second), and C_L rises with it. The ball turns more sharply the slower it goes.

This is the “late bite” keepers talk about. Fans remember Roberto Carlos’s free kick against France in 1997, which seemed to be heading well wide before it swung back into the goal.