Where is he, how fast, how hard? From position to acceleration, and back again

Contents

• The football question
• The concept
• Back again needs a starting point
• Down the chain, noise gets louder
• What coaches want to know
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

A winger starts almost from a standstill, chases a ball into space, accelerates hard, and gets close to top speed. What can calculus tell us about one run like that?

Quite a lot, it turns out. Take it apart one question at a time.

• Where is he? That's position.
• How quickly is that position changing? That gives us velocity: speed, with a direction.
• How quickly is that velocity changing? That gives us acceleration, the second derivative of position.
• How is the acceleration itself changing? That's the difference between a smooth build-up and an explosive burst.

Each step is a derivative of the one before. And each can be undone by integrating.

The concept

Here's the whole chain for our winger's three-second run (made-up example numbers, from rate of change and the parts that followed):

Going down the chain:

• Position to speed: the derivative, how quickly position is changing.
• Speed to acceleration: the derivative again.
• Acceleration to change in acceleration: the second derivative of speed.

Going back up:

• Speed to position: integration, the area under the speed graph. 2 + 5 + 7.5 metres takes him from 0 to 14.5.
• Acceleration to speed: integrate again. Add up each second's acceleration and you get back to his speed.

Position → velocity → acceleration, and back again. That's calculus on a football pitch.

Back again needs a starting point

There's a catch in going back up. Suppose all we're told is his acceleration: 3, 3, then 2 m/s². Add it up and his speed rises by 3, then 3, then 2. But from what?

If he started at 0.5 m/s, his speed goes 0.5, 3.5, 6.5, 8.5, and he covers 14.5 m. If he started from a dead standstill, it goes 0, 3, 6, 8, and he covers 13 m. Same acceleration, a metre and a half less ground.

$$\int a(t)\,dt = v(t) + C$$

The same happens one step further up: speed tells you how far he ran, but not where on the pitch he started. That's why tracking systems record positions, the top of the chain, and work everything else out from there.

Down the chain, noise gets louder

Modern tracking data gives us player positions through time. From that we can derive velocity, acceleration and much more. But there's a price for going down the chain.

Say each position reading is off by about 10 cm, a made-up figure for illustration, and the system takes ten readings a second. Work out speed from the change between two readings, then acceleration from the change between two speeds. A simulation of our smooth sprint from the derivative, run 2,000 times, gives:

Each derivative divides a small difference by a tiny time, and the errors get divided along with it. One derivative makes speed usable but rough. Two make raw acceleration pure noise. That's why tracking providers smooth positions before they take any derivatives.

Going up the chain does the opposite. Add up those same noisy speeds over the five-second run and the errors largely cancel: the distance comes out within about 0.14 m of the truth. Derivatives amplify noise; integrals smooth it.

What coaches want to know

Calculus gets taught as a collection of strange formulas, but really it's a way of describing movement and change, and football is full of both:

• Runs into space.
• Recovery sprints.
• The press.
• An overlapping full-back.
• A defender trying to close a gap.
• A striker trying to lose him.

Suddenly "he ran 10 kilometres" isn't nearly enough. Coaches and analysts want to know:

• Where did he run? Position.
• How quickly did he get there? Speed.
• How hard did he accelerate? Acceleration.
• How often did he repeat those bursts? Counting peaks in acceleration.
• How did that drop off as the match went on? Comparing the whole chain early and late.

Why it matters

• One chain explains the lot. Distance covered, top speed, sprints, accelerations, fatigue: every tracking metric sits somewhere on it.
• Direction matters. Going down the chain magnifies errors, going up smooths them, which is why raw acceleration figures need careful handling.
• You need a starting point. Integrating back up only works if you know where he started and how fast he was going.
• That's why the maths matters to coaches, analysts and sports scientists. A player's movement can be broken down, measured, compared and tracked over time, and then built back up again.

Limitations

• Real runs aren't straight lines. Position and velocity on a pitch have two directions, across and along, and turning is acceleration too; player movement as vectors handles both.
• Smoothing has a cost. It removes noise but also blunts the sharpest real changes, so the true peak acceleration can be understated.
• The 10 cm error is illustrative. Real systems differ; the point is how errors grow with each derivative, not the exact figure.

And if you've followed this far, you probably understand a fair bit more calculus than you thought you did. Now I just need a branch of mathematics that can explain Celtic's last defeat.

Try it yourself

Pick a moment in a match replay: a full-back overlapping, say. Describe it three ways: where he went, how fast he was going, and how hard he accelerated. Which of the three would you notice watching live, and which only shows up in the data?

Further reading

• Integration and the fundamental theorem of calculus, 3Blue1Brown. Why derivatives and integrals undo each other.
• Fundamental theorem of calculus, Wikipedia. The formal statement of "and back again".
• Numerical differentiation, Wikipedia. Estimating derivatives from data, and why errors grow when the steps get small.