This post is merely to provide a link to a video of a mathematical machine that I made over the Christmas holidays. Even if maths isn’t your thing, you might enjoy the movements and the rhythms.

# Month: January 2022

## Walk-run strategies in running races

If you’re running an endurance race such as a 10K, half-marathon, or marathon, it might seem obvious that the quickest way of getting to the finishing line is to run continuously over the entire distance. But some people (notably Jeff Galloway) suggest that, particularly if you’re a slower runner, you might actually finish sooner if you walk some of the way. The rest that walking gives you can boost your running pace enough to make your overall pace faster. Galloway claims gains of up to 7 minutes in a half-marathon and 13 minutes in a marathon.

Websites such as Galloway’s give list of suggested run/walk ratios, but I haven’t found anything that lets you see what overall pace you’ll do if you follow a given run-walk strategy. Here, I aim to fill that gap.

The table below tells you how long (in time) your bursts of running need to be to achieve a given overall average pace (left-hand column), for different paces of running (top row). It assumes that you are going to alternate bursts of running with 1-minute walking breaks, and that you walk at a pace of 15 minutes per mile.

Here are two ways in which you might use the table.

*Example 1: *Suppose that you aspire to run the race at an average 10 minutes per mile. How fast and how long do your running bursts need to be? Locate 10:00 in the left-hand column of the table. Now reading across, you come to the time 0:47. Looking at the top of the column, the pace in bold is 7:00 minutes per mile. So you can achieve a 10:00 per mile pace by running at 7:00 per mile for bursts of 47 seconds, and walking for a minute between them. Proceeding along the same row, you could get the same overall pace by running at 7:30 per mile for bursts of 1m00s, and so on up to a probably more realistic 9:30 per mile for bursts of 6m20s.

*Example 2:* Suppose that you think you can run at 9:30 per mile while you are actually running. What is the average pace for different lengths of running burst? Locate 9:30 along the top of the table. Going down the column to where it says “6:20”, and reading across to the left-hand column, you find that running bursts 6m20s long will give you an average pace of 10:00 per mile. Similarly, running bursts 2m51s long will give you an average pace of 10:30, and so on.

Note that the table says nothing about what you are capable of. It just tells you what your overall pace will be if you can achieve certain durations and paces for the running bursts.

If you want walking breaks longer than 1 minute, increase the length of the running bursts in the same proportion.

If the combination you want isn’t in the table, or you want to assume a different walking pace, or you want to work in kilometres, there’s a formula below that you can use.

#### The formula

Let your walking and running paces be and respectively. You can specify these in either minutes per mile or minutes per kilometre. You’ll need to convert min:sec values to decimal values of minutes.

Let the durations of the walking breaks and running bursts be and respectively. You can use any units of times for these durations (as long as it’s the same for both).

Your average pace (in the same units that you used to specify your running and walking paces) is given by

If you’re walking for 1-minute breaks, this simplifies to

#### Derivation

Runners usually express how fast they are running in terms of minutes per mile (or kilometre). I’m going to call this the running *pace*. However, because we want to average over time, rather than distance, we need to do the averaging using *speeds* expressed as miles (or kilometres) per minute.

Let your walking and running paces be and respectively. The corresponding speeds are and .

We will assume that you alternate running for walk for a time and then running for a time .

We’re going to work out the weighted average of your walking and running paces to work out the overall average pace. Because we’re averaging over time, not distance, we need to do the averaging using speeds, not paces, and then convert back to a pace.

If your time-average pace is , your time-average speed (distance-per-time) is and is given by

so your time-average pace (time-per-distance) is

which we can tidy up a bit to give

Where = 1, this simplifies to

#### Acknowledgement

Many thanks to Graham Rose for his wonderful cartoon. It feels like that from inside, too.

Mathematical typesetting was done using the QuickLatex plugin.