How quickly is he getting up to speed? Rate of change, or, yes… calculus
Every time you say a winger shot away from a defender, you're talking about rate of change. That's the heart of calculus, and a few seconds of sprinting are enough to see the derivative at work.
Beginner Part 1 of Calculus Through Football
Contents
The football question
Mention calculus and quite a few football people immediately switch off and are heading out the room. But if you watch football, you already use some of the thinking behind calculus all the time:
- "He was up to speed incredibly quickly."
- "The defender couldn't accelerate fast enough to catch him."
- "Look how quickly he's closing him down."
- "He's lost a yard of pace."
Every one of those is about how quickly something is changing. So how do you put a number on it?
The concept
A rate of change is how quickly one thing changes as another changes. Speed is the rate of change of position: metres per second. Acceleration is the rate of change of speed: how many metres per second faster he gets, every second.
The simplest way to measure it is to take two moments and compare:
$$\text{Rate of change} = \frac{\text{change in speed}}{\text{change in time}}$$
In plain football
- Change in speed: how much quicker he's going at the end than at the start.
- Change in time: how long it took.
- Divide one by the other and you get acceleration, in metres per second, per second: m/s².
A football example
A winger picks up the ball and sets off. Here's his speed, second by second (made-up example numbers):
| Time | Speed |
|---|---|
| 0 seconds | 0.5 m/s |
| 1 second | 3.5 m/s |
| 2 seconds | 6.5 m/s |
| 3 seconds | 8.5 m/s |
Over the whole three seconds:
$$\frac{8.5 - 0.5}{3 - 0} = \frac{8}{3} \approx 2.67 \text{ m/s}^2$$
In plain football
- 8.5 − 0.5 = 8: he's 8 metres per second quicker than when he started.
- 3 − 0 = 3: it took him three seconds.
- 2.67 m/s²: his speed increased by an average of roughly 2.67 metres per second, every second.
But that's an average. Break it down second by second and a different picture appears:
| Interval | Change in speed | Acceleration |
|---|---|---|
| 0 to 1 s | +3.0 m/s | 3.0 m/s² |
| 1 to 2 s | +3.0 m/s | 3.0 m/s² |
| 2 to 3 s | +2.0 m/s | 2.0 m/s² |
He accelerates hard for two seconds, then starts to level off as he nears top speed. Plot speed against time and you can see it: the steeper the line, the faster his speed is changing.
At this exact moment: the derivative
An average over a second is still an average. The question calculus really asks is: how quickly is his speed changing at this exact moment? That's the derivative.
The trick is to shrink the gap. Take a smooth sprint, where speed rises quickly at first and then levels off, as real sprints do (made-up example numbers: a top speed of 9.5 m/s). Measure the average acceleration from the one-second mark over smaller and smaller gaps:
| Gap after 1 second | Average acceleration |
|---|---|
| 1 second | 2.33 m/s² |
| 0.5 seconds | 2.81 m/s² |
| 0.1 seconds | 3.30 m/s² |
| 0.01 seconds | 3.43 m/s² |
| 0.001 seconds | 3.44 m/s² |
As the gap shrinks, the answer settles on one number: 3.44 m/s². That's his acceleration at exactly one second, the derivative of speed at that instant. On the graph, it's the steepness of the curve at that single point.
$$a(t) = \lim_{h \to 0} \frac{v(t + h) - v(t)}{h}$$
In plain football
- v(t) is his speed at time t; v(t + h) is his speed a moment h later.
- The fraction is the same "change in speed ÷ change in time" as before, over a tiny gap.
- lim, h → 0 means: keep shrinking the gap and see what number the answer settles on. That number is the derivative, a(t), his acceleration at that exact moment.
GPS vests do a version of this every match: they record a player's position many times a second, and turn the tiny changes into speed, and the tiny changes in speed into acceleration.
Going the other way
Calculus works in both directions. The derivative turns speed into acceleration. Its opposite, the integral, turns speed back into distance: the area under the speed graph. That's the subject of integration.
Why it matters
- Rate of change is everywhere in football data. Speed, acceleration, how quickly a team's xG is rising through a match, how fast a player's output is declining with age.
- Averages hide the moment. A three-second average of 2.67 m/s² hides the fact that he was accelerating at 3 for two seconds and then easing off.
- Acceleration often matters more than top speed. Many sprints in a match are short, and the player who gets to 7 m/s first can win the ball even if the other is quicker over 40 metres.
- Machine learning runs on it. Models learn by working out how quickly their error changes as they adjust each setting, and moving downhill. That's the derivative again: see gradient descent.
Limitations
- Real data is noisy. GPS positions jump about a little, and dividing tiny changes by tiny times magnifies that noise. Tracking providers smooth the data first.
- Straight lines are an approximation. Treating speed as changing steadily within each second is close for a sprint, but not exact.
- Speed isn't the whole story. Direction changes, decelerating and turning are rates of change too, and often decide duels.
So the next time you're watching football and say "he absolutely shot away from him", you might not realise it, but you're already thinking about rate of change, acceleration and calculus.
Try it yourself
Next time you watch a match, pick a winger's sprint and count the seconds from when he sets off to when he's at full pelt. If he reaches around 9 m/s (roughly 32 km/h) in 3 seconds from a standing start, what's his average acceleration? Would a defender who needs 4 seconds to reach the same speed ever catch him?
Further reading
- The paradox of the derivative, 3Blue1Brown. How a rate at "an exact moment" makes sense, from the Essence of calculus series.
- Derivative, Wikipedia. The definition as a limit, with the tangent-line picture.
- Acceleration, Wikipedia. Acceleration as the rate of change of velocity, and its units.