Still accelerating, or at full speed? The derivative

Contents

• The football question
• The concept
• What the commentary means in derivatives
• A football example
• The questions coaches ask
• A yard of pace
• Why it matters
• Limitations
• Try it yourself
• Further reading

The football question

A winger is one and a half seconds into a sprint. Is he still accelerating? How quickly, right now?

Commentators answer questions like that all the time:

• "He's still accelerating."
• "He's reached full speed."
• "He's beginning to slow down."
• "The defender is closing the gap."

All of those statements are really about how something is changing at a particular moment. Putting a number on "right now" is exactly what a derivative does.

The concept

Here's our winger's speed, second by second (made-up example numbers, from rate of change):

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

Suppose we want to know what is happening at exactly 1.5 seconds. We zoom in on that tiny moment. On a speed-versus-time graph, we can draw a line touching the curve at that point: a tangent. The slope of that line tells us the player's instantaneous rate of change, his acceleration at that exact moment.

That is essentially what a derivative tells us. It's written

$$a(t) = \frac{dv}{dt}$$

In our table, speed rises steadily between 1 and 2 seconds, so the tangent at 1.5 seconds is just that second's straight line: 3.0 m/s². Real sprints aren't made of straight lines, though. Speed climbs quickly at first and then levels off in a smooth curve, and the tangent's slope changes at every moment.

What the commentary means in derivatives

Commentary	The derivative
"He's still accelerating"	Acceleration is above zero: speed is still rising
"He's reached full speed"	Acceleration has fallen to about zero: the speed curve is flat
"He's beginning to slow down"	Acceleration is below zero: speed is falling
"The defender is closing the gap"	The distance between them is shrinking: its rate of change is negative

A football example

Take a smooth sprint from a standing start that levels off at a top speed of 9.5 m/s (made-up example numbers, using the kind of curve sports scientists fit to real sprint data). Draw the tangent at three moments:

Do that at every moment and the derivative becomes a curve of its own: his acceleration profile.

Time	Speed	Acceleration
0 s	0.0 m/s	7.9 m/s²
0.5 s	3.2 m/s	5.2 m/s²
1 s	5.4 m/s	3.4 m/s²
1.5 s	6.8 m/s	2.3 m/s²
2 s	7.7 m/s	1.5 m/s²
3 s	8.7 m/s	0.6 m/s²

At 1.5 seconds he's still accelerating, at 2.3 m/s², but less than a third as hard as on his first step.

The questions coaches ask

The acceleration profile answers the questions coaches actually care about:

• How quickly do they become quick? His first step is his hardest: 7.9 m/s² from a standing start.
• How long does it take them to reach maximum speed? Strictly, he never quite gets there; the curve keeps edging closer. He reaches 95% of it, 9 m/s, after about 3.6 seconds.
• At what point does their acceleration begin to fall? Straight away. In this sprint it halves roughly every 0.83 seconds: 7.9, then about 4, then about 2.
• Can they repeat that acceleration late in a match?
• Is their acceleration curve improving through training?

The last two are the same question: compare the curve now with the curve before.

A yard of pace

Say that late in a match, tired, our winger's acceleration fades a little faster: he takes slightly longer to reach the same top speed (made-up example numbers). His first step drops from 7.9 m/s² to 6.8.

Top speed hasn't changed. But after two seconds of sprinting, the tired version of him has covered 8.9 m against 9.8 m fresh: 0.86 m behind. A yard is 0.91 m. That's "he's lost a yard of pace", measured.

Show the mathsThe sprint curve, its derivative, and the distance run. Optional.

The speed curve used here is

$$v(t) = v_{\max}\left(1 - e^{-t/\tau}\right)$$

with top speed \(v_{\max} = 9.5\) m/s and \(\tau = 1.2\) s fresh (1.4 s tired). Its derivative is the acceleration:

$$a(t) = \frac{dv}{dt} = \frac{v_{\max}}{\tau}\, e^{-t/\tau}$$

so \(a(0) = 9.5 / 1.2 \approx 7.9\) m/s², and acceleration halves every \(\tau \ln 2 \approx 0.83\) s. Speed reaches 95% of top speed when \(e^{-t/\tau} = 0.05\), at \(t = \tau \ln 20 \approx 3.6\) s.

Going the other way, the distance run is the integral of speed:

$$x(t) = v_{\max}\left(t - \tau\left(1 - e^{-t/\tau}\right)\right)$$

giving \(x(2) \approx 9.75\) m fresh and \(8.89\) m tired.

Why it matters

• "Right now" is what matters in a duel. Whether a defender can catch a winger depends on their accelerations in the next second, not their averages.
• Top speed isn't the whole story. Two players with the same top speed can have very different acceleration profiles, and the quicker starter wins most short races.
• It's measurable. GPS and tracking data record speed many times a second, so a player's acceleration curve can be tracked through a match and across a season.
• It's trainable. Derivatives don't just tell us where a player is. They help tell us how their performance is changing, moment by moment. And that gives coaches something they can actually train, measure and improve.

Limitations

• The sprint curve is a model. Real sprints are close to it from a standing start, but players rarely start from a standstill in a match; they're often already moving.
• Measured acceleration is noisy. Small errors in position become big errors in acceleration, so tracking data is smoothed before derivatives are taken.
• Straight lines aren't sprints. Curved runs, changes of direction and decelerating to stop are all derivatives too, and all matter.

Try it yourself

Watch a replay of a winger beating a full-back and pause it every half-second. Is the gap between them growing faster, growing slower, or shrinking? You've just estimated the derivative of the distance between them.

Further reading

• Limits and the definition of derivatives, 3Blue1Brown. How the tangent's slope comes from shrinking gaps, from the Essence of calculus series.
• Tangent, Wikipedia. The line that touches a curve at one point, and its slope as the derivative.
• A spreadsheet for sprint acceleration force-velocity-power profiling, JB Morin. How sports scientists fit this kind of speed curve to real sprints, with a free spreadsheet.