# Is his acceleration fading? The second derivative

Source: https://www.footballdatascience.co.uk/learn/second-derivative
Published: 2026-09-17

> 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.

## 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](/learn/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](/learn/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:

$$\begin{aligned} &\text{change in acceleration} \\ &= \frac{\text{acceleration now} - \text{before}}{\text{time between}} \end{aligned}$$

<div class="plain" markdown="1">
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**.
</div>

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

$$\frac{d^2 v}{dt^2} = \frac{d}{dt}\left(\frac{dv}{dt}\right)$$

<div class="plain" markdown="1">
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.
</div>

<div class="plain" markdown="1">
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.
</div>

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):

<figure class="rank-chart">
<div role="img" aria-label="Bar chart of acceleration each second. Player A: 3, 3, 2, then none once at top speed. Player B: 3, 2, 1.5, 1, 0.5.">

</div>
<figcaption>The same first second. A holds 3 m/s² for another second before easing off; B's acceleration starts dropping straight away. The step down from one bar to the next is the second derivative.</figcaption>
</figure>

Both reach 8.5 m/s in the end: A after three seconds, B after five.

<figure class="rank-chart">
<div role="img" aria-label="Speed against time. Both players start at 0.5 metres per second and reach 3.5 after one second. A reaches 8.5 at three seconds; B reaches 8.5 at five seconds.">

</div>
<figcaption>A's speed line runs straight for two seconds, then bends. B's starts bending after one second. By the time both are at top speed, A is 3 metres ahead.</figcaption>
</figure>

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](https://www.3blue1brown.com/lessons/higher-order-derivatives), 3Blue1Brown. The second derivative as the bend in a graph, with a motion example.
- [Second derivative](https://en.wikipedia.org/wiki/Second_derivative), Wikipedia. Notation, concavity and what the sign tells you.
- [Jerk (physics)](https://en.wikipedia.org/wiki/Jerk_(physics)), Wikipedia. The rate of change of acceleration, its units, and where it matters.