Is his acceleration fading? The second derivative
Two players can hit the same top speed and look identical for a second, then one pulls clear. The difference is how quickly their acceleration fades, and that is the second derivative.
Beginner Part 3 of Calculus Through Football
Contents
The football question
A winger chases a ball into space, going from almost standing still to 8.5 m/s in three seconds (made-up example numbers, from rate of change). He's still getting faster. But is the burst starting to fade?
Break his acceleration down second by second:
| Interval | Acceleration |
|---|---|
| 0 to 1 s | 3.0 m/s² |
| 1 to 2 s | 3.0 m/s² |
| 2 to 3 s | 2.0 m/s² |
He's still getting faster, but the rate at which he's getting faster is beginning to fall. That's the second derivative.
The concept
The derivative asks how quickly something is changing. The second derivative asks how quickly that change is changing. Written down it's horrible: the rate of change of the rate of change.
For our winger, take the change in acceleration from one second to the next:
$$\begin{aligned} &\text{change in acceleration} \\ &= \frac{\text{acceleration now} - \text{before}}{\text{time between}} \end{aligned}$$
In plain football
- First to second: 3.0 to 3.0, no change at all. He's accelerating just as hard.
- Second to third: 3.0 to 2.0, a drop of 1.0 m/s² in a second. The burst is fading as he gets near top speed.
- The units stack up: metres per second, per second, per second, m/s³.
Physicists have a name for it when it's applied to motion, by the way. They call it jerk. No kidding.
In calculus notation, the second derivative of speed is written
$$\frac{d^2 v}{dt^2} = \frac{d}{dt}\left(\frac{dv}{dt}\right)$$
In plain football
- dv/dt is his acceleration: the first derivative of speed.
- d/dt of that is how quickly his acceleration is changing: the second derivative of speed.
- Negative means the burst is fading. Zero means he's holding it. Positive means he's still building up.
A note on names
Count from position instead of speed and everything shifts by one. Speed is the first derivative of position, acceleration the second, and jerk the third. So "the second derivative" of a sprint can mean acceleration or jerk, depending on where you start counting. Here we start from speed, so it's jerk.
On a speed–time graph, the second derivative is the bend. While it's zero the speed line runs straight; when it turns negative, the line starts to bend over and flatten.
A football example
Two players can both hit 8.5 m/s. They can even look identical over the first second. But if one of them holds that acceleration for longer, he's a completely different athlete.
Player A is our winger. Player B starts exactly the same, then his acceleration fades straight away (made-up example numbers):
Both reach 8.5 m/s in the end: A after three seconds, B after five.
How far apart does that put them?
| Distance | Player A | Player B | B is behind by |
|---|---|---|---|
| 5 m | 1.67 s | 1.71 s | 0.05 s |
| 10 m | 2.43 s | 2.59 s | 0.16 s |
| 20 m | 3.65 s | 3.97 s | 0.32 s |
| 30 m | 4.82 s | 5.18 s | 0.35 s |
Over the first five metres there's almost nothing in it. The difference shows up once the run goes on: from five seconds, when both are at top speed, A is 3 metres ahead and stays there. Same first step, same top speed, a different second derivative.
What fans already say
- "He keeps accelerating away from him." A is holding his acceleration; the second derivative is near zero.
- "He had the initial burst but couldn't sustain it." A fast first second, then a steeply negative second derivative.
- "The defender caught him in the end." The attacker's acceleration faded sooner than the defender's.
Different positions, different curves
| Position | What matters |
|---|---|
| Winger | Explosive over the first five metres, and able to hold it into space |
| Full-back | Producing that burst again and again for ninety minutes |
| Centre-back | May never hit the same top speed, but a sharp first few metres can be the difference between a block and a goal |
The table above adds one thing. Over five metres, how long a player holds his acceleration barely matters: the first second does most of the work. Over 20 or 30 metres it's worth a third of a second. So the centre-back's race is mostly about the first derivative, the winger's and full-back's about the second.
Why it matters
- Top speed hides the story. A and B have the same top speed, and a scout looking only at that would call them equal.
- The shape of the run is trainable. How long a player holds his acceleration can be measured from tracking data and worked on in training.
- It spots fatigue. A full-back whose acceleration fades sooner in the 80th minute than the 10th is tiring, even if he still reaches the same top speed.
- We've moved on from "how fast is he?" to "how is his speed changing while the run unfolds?" That's the second derivative, and it's where calculus starts earning its keep.
Limitations
- One-second steps are coarse. Real acceleration changes smoothly; tracking data samples it many times a second.
- Second derivatives amplify noise. Each derivative magnifies small measurement errors, so tracking providers smooth heavily before calculating jerk.
- These are made-up players. The numbers show the idea; real profiles vary with starting speed, direction and the state of the pitch.
Try it yourself
Watch a long run in a match replay, a counter-attack from one box to the other. Does the runner keep pulling away from the chaser, hold the gap, or start to be reeled in? Which of those is a second derivative near zero, and which a negative one?
Further reading
- Higher order derivatives, 3Blue1Brown. The second derivative as the bend in a graph, with a motion example.
- Second derivative, Wikipedia. Notation, concavity and what the sign tells you.
- Jerk (physics), Wikipedia. The rate of change of acceleration, its units, and where it matters.