# Running: what should be my speed for a given ETA?

# The speed = f(race time) equation

Below are the different expressions of the equation that binds average speed and ETA (race time).##

*Légend* :##
**d** : distance ; **t** : time ; **v** : velocity (speed)##
**d _{km}** : distance, expressed in kilometers##

**t**: time, expressed in minutes##

_{min}**v**: speed, in kilometers per hour.

_{km/h}*Example : 43 minutes and 20 seconds, don’t use 43.20 in the equation, but rather 43 + 20/60, so 43.3333.*

## Example of a numerical evaluation to calculate the necessary average speed

Example of a 10 km run, thus **d _{km}**=10 ; you’re aiming at 50 minutes, thus

**t**= 50 ; the equation supplies

_{min}**v**= 60*10/50 = 12 km/h. Meaning you will have to run at an average speed of 12 km/h to cover the distance in 50 minutes.

_{km/h}# Pace

The Pace, that I note **p** herein, is the reverse expression of the speed. It means that it represents the time needed to run a unit of distance. It is usually expressed in *minutes per kilometer*, or in minutes per mile in some English-speaking countries (e.g. the USA).

Smartphone apps and an increasing number of smartwatches display the pace while you’re running. The lower the pace the faster you’re running, and vice-versa.## E.g. if you ran 10 km in one hour, then your pace was 60 minutes / 10 km = 6:00 minutes per kilometer.

# Pace : speed conversion

## The equations

Let’s note p the pace, and v the speed (velocity), thefore we have the equivalence:##

### Practical application

Let’s consider a speed of v = 14.5 km/h, what is the corresponding pace?

Answer:####
p = 3600 / 14.5 = 248 seconds, (we neglected the decimal part)##
which is 4 minutes (4 x 60 = 240 seconds) and 8 seconds, simply noted **4:08**.

What about a pace of 5:25 (minutes per km), what is the corresponding speed?

Answer:####
v = 3600 / (5 x 60 + 25), therefore **11.08 km/h** (rounded to the second decimal).

# Running time : pace conversion

## The formula

Let p_{s} be the pace in **seconds per km**, let t be the running time in seconds, and d the distance run in meters. Therefore we have the following equivalence :##

### Practical application

Let’s consider a running time of t = 1:42:47 over a half-marathon. What is the corresponding pace?

Answer:####
*Always exploit the equations given using International System units: meters and seconds!##
Convert the running time in seconds : t = 1x3600 + 42x60 + 47 = 6167 seconds,##
Convert the distance in meters; a half-marathon is 21,097 m .##
Finally, apply the above formula p = t/d, i.e. p = 1000 x 6167 / 21097 = 292 seconds per km. That we will now convert into minutes and seconds, which are 4 minutes and 52 secondes, simply noted **4:52** .

### How did the guy do to convert seconds into minutes + seconds?

An easy way to convert 292 seconds into minutes, mentally, is to look for the closest multiple of 60.## We know that 5 times 60 equals 300, so the pace will be very close to 5:00 per km. But since 292 is only 8 seconds less than this multiple, we take them off, giving 5:00 - 0:08 = 4:52/km .

# Reference charts

I’ve drawn up these charts for your convenience, with classical distances and running times.## The values in the charts are the average speeds in km/h that you need to run at for the corresponding ETAs.##

**21 097 m**is a half-marathon’s distance,##

**42 195 m**is a marathon’s.

## Speed chart, in km/h

_{distance} / ^{time} |
20 min | 25 min | 30 min | 35 min | 40 min | 45 min | 50 min | 55 min | 1:00 | 1:05 | 1:10 | 1:15 | 1:20 | 1:25 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

5 km |
15.00 | 12.00 | 10.00 | 8.57 | 7.50 | 6.67 | 6.00 | |||||||

10 km |
20.00 | 17.14 | 15.00 | 13.33 | 12.00 | 10.91 | 10.00 | 9.23 | 8.57 | 8.00 | 7.50 | 7.06 | ||

20 km |
21.82 | 20.00 | 18.46 | 17.14 | 16.00 | 15.00 | 14.12 | |||||||

half-marathon |
23.01 | 21.10 | 19.47 | 18.08 | 16.88 | 15.82 | 14.89 |

### (continued)

_{distance} / ^{time} |
1:30 | 1:35 | 1:40 | 1:45 | 1:50 | 1:55 | 2:00 | 2:05 | 2:10 | 2:15 | 2:20 | 2:25 | 2:30 | 3:00 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

10 km |
6.67 | 6.32 | 6.00 | 5.71 | 5.45 | 5.22 | 5.00 | |||||||

20 km |
13.33 | 12.63 | 12.00 | 11.43 | 10.91 | 10.43 | 10.00 | 9.60 | 9.23 | 8.57 | 8.00 | 7.50 | 7.06 | 6.67 |

half-marathon |
14.06 | 13.32 | 12.66 | 12.06 | 11.51 | 11.01 | 10.55 | 10.13 | 9.74 | 9.04 | 8.44 | 7.91 | 7.45 | 7.03 |

marathon |
21.10 | 20.25 | 19.47 | 18.08 | 16.88 | 15.82 | 14.89 | 14.07 |

### (continued)

_{distance} / ^{time} |
3:10 | 3:20 | 3:30 | 3:40 | 3:50 | 4:00 | 4:20 | 4:40 | 5:00 | 5:20 | 5:40 | 6:00 | 6:30 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

5 km |
|||||||||||||

10 km |
|||||||||||||

20 km |
6.32 | 6.00 | 5.71 | 5.45 | 5.22 | 5.00 | |||||||

half-marathon |
6.66 | 6.33 | 6.03 | 5.75 | 5.50 | 5.27 | |||||||

marathon |
13.32 | 12.66 | 12.06 | 11.51 | 11.01 | 10.55 | 9.74 | 9.04 | 8.44 | 7.91 | 7.45 | 7.03 | 6.49 |

## pace/speed conversion chart, time per kilometre.

pace (min/km) | 2:45 | 2:50 | 2:55 | 3:00 | 3:05 | 3:10 | 3:15 | 3:20 | 3:25 | 3:30 | 3:35 | 3:40 | 3:45 | 3:50 | 3:55 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

speed (km/h) | 21.82 | 21.18 | 20.57 | 20.00 | 19.46 | 18.95 | 18.46 | 18.00 | 17.56 | 17.14 | 16.74 | 16.36 | 16.00 | 15.65 | 15.32 |

### (continued)

pace (min/km) | 4:00 | 4:05 | 4:10 | 4:15 | 4:20 | 4:25 | 4:30 | 4:35 | 4:40 | 4:45 | 4:50 | 4:55 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

speed (km/h) | 15.00 | 14.69 | 14.40 | 14.12 | 13.85 | 13.58 | 13.33 | 13.09 | 12.86 | 12.63 | 12.41 | 12.20 |

### (continued)

pace (min/km) | 5:00 | 5:05 | 5:10 | 5:15 | 5:20 | 5:25 | 5:30 | 5:35 | 5:40 | 5:45 | 5:50 | 5:55 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

speed (km/h) | 12.00 | 11.80 | 11.61 | 11.43 | 11.25 | 11.08 | 10.91 | 10.75 | 10.59 | 10.43 | 10.29 | 10.14 |

### (continued)

pace (min/km) | 6:00 | 6:15 | 6:30 | 6:45 | 7:00 | 7:15 | 7:30 |
---|---|---|---|---|---|---|---|

speed (km/h) | 10.00 | 9.60 | 9.23 | 8.89 | 8.57 | 8.27 | 8.00 |

*All right, I’ve gone down to 7:30/km because it was the aerobic running speed of my younger son over 4 km when he was five . . . don’t feel bad if this is your running speed! What is more important is to progress, or even at least to run! So many can’t!*

# Practical application: the possible race strategies to reach your goal

Start your race at the same speed as the average speed supplied by the reference chart. Or . . . start slower, then accelerate over the last kilometers to increase it, until your average speed equals the one given in the charts.## This latter method ensures a good warm up, then a steady and controlled cruise speed, and finally will allow you to gauge the amount of risk you can take towards the end if you tap in to your reserves, to grab a few seconds or minutes. However, don’t accelerate too late towards the end, especially during long races, as your average speed will barely vary . . .

# What if I want to know my ETA for a given speed?

It is the reciprocal problem to the initial topic’s question. The speed-time equation allows you to swap speed and time.

## Example of a numerical evaluation for an ETA

Example of a half-marathon, thus **d _{km}**=21.1 ; you’re thinking of running at 10.5 km/h, what would your ETA be?##
The equation gives

**t**= 60 x 21.1 / 10.5 =

_{minutes}**120.57**. Therefore 2 hours and 0.57 minutes, wait, I didn’t say 57 seconds! So you have to multiply by 60 to get seconds, so 0.57 x 60 = 34 secondes, roughly. Thus a total of 02:00:34. Good luck!