In a recent Marilyn vos Savant column a person describes a trip made by two people, one walking and one on a bicycle. They start together, one walking and one riding. After the rider goes some distance ahead of the walker the bike is parked and the (ex-)rider starts walking. When the (original) walker catches up to the parked bike he/she takes the bike and starts riding, passing the walker and parking some distance beyond. They continue alternating walking/riding in this manner to their destination.
The question raised is whether this pattern results in a faster transit than simply walking the whole distance, sans bike. One position is that, since someone is always walking, the time to cover the distance is the same as sans bike. Marilyn’s position is that since neither person walked the entire distance the average speed of both was increased, thus the transit was shorter, i.e., took less time.
I think Marilyn is right but I don’t particularly like her explanation. Anyone want to take a shot at ‘splaining why she’s right?
Side note: There are actual races IRL with the bicycle replaced with a horse and walkers replaced with runners. This has absolutely nothing to do with my question.
Also, the first person who uses the word “treadmill” will be taken outside and shot.
Take an example. Let’s say you can walk 1 mph and ride a bike 2 mph. The trip is 4 miles long. After one hour Person A is walking and at the 1 mile mark, Person B is riding and at the 2 mile mark, leaves the bike.
At two hours Person A is at 3 mile mark and Person B is at 2 mile mark and gets on the bike.
At three hours they are both at the 4 mile mark.
This would be true if the person on the bike stopped and waited for the walker to arrive before dismounting and starting to walk. That is not the case though and the bike rider gets a “head start” on his walking until the original walker reaches the bicycle.
This example shows how it works; to be clear, only walking would take four hours, so this is one hour less.
In effect it doesn’t matter how many people there are, as long as one person can use the bike for any part of the journey (even 10 minutes), he/she will be quicker than if they have to walk the whole way. If there is two people and both use the bike for some time (no matter how short) there will be a point where they meet up at the destination that is sooner than when one of them had walked the whole way.
You can simplify the problem by considering a 2-mile-long course with just two stages: person A rides the first mile, parks the bicycle and walks the rest; B does the opposite.
If they walk at, say, a 10 min/mile pace and ride at 4 min/mile, they both arrive after 14 minutes, which is 6 minutes sooner than required to walk the entire distance.
What’s wrong with the “Someone is always walking” argument? Simple: there’s plenty of time when both are walking (in the example above, minutes 4 through 10), during which walked distance is being accumulated at twice the single-person rate.
There is also some argument that riding the bike is less tiring (you can coast occasionally) and/or uses different muscle groups, thereby helping you maintain a faster pace when walking. In the man+horse races mentioned in the OP, the optimum strategy seems to be to station the horse at the point where the runner begins to tire. The runner can then get on the horse and rest (somewhat) while still covering ground. The horse gets to rest while waiting for the next runner to catch up, so everyone stays fresher. The same theory would apply to the walkers+bike, except the bike doesn’t get tired.
Hijack: even if the biker did have to stop and wait for the walker to catch up, they might get to their destination fast since they will be less tired since they will spend half their time waiting around rather than walking.
ETA: that wasn’t meant as a response to Stana’s post although that has merit too
Take the limiting case - where the bicycle has infinite speed. There is always someone walking, indeed there are always two people walking. But they get there twice as fast.
Actually, this has absolutely everything to do with your question.
The so-called “ride and tie” system that you’re describing, which has now given its name to a specific type of equestrian/pedestrian competitive event, originated back in the distant past with horses rather than bicycles.
Our ancestors developed the “ride and tie” method for two people using one horse precisely because it does get both people to their destination in a shorter time than the alternatives of one or both people walking the whole way.
(I’ve done “ride and tie” with a bicycle, not in competitive racing but purely for practical efficiency in moving two people with one bicycle from Point A to Point B. The typical approach to pacing is for the rider to get off and start walking just before s/he gets out of sight of the following walker. That way, both people are at least potentially in sight of each other and of the bicycle at all times, reducing the chances of (a) someone getting lost and (b) the bicycle getting stolen.)
But if they didn’t have a bicycle at all, there would always be two people walking.
The flaw in the “Someone is always walking” argument is that once they arrive, there isn’t someone walking anymore. To use jonesj2205’s example, with two people walking, it takes four hours. When using the bike, there’s someone always walking only for the first three hours. After that, for the fourth hour, there isn’t anyone walking anymore, because they are both there already.
Just to be nitpickingly precise, the ‘ride and tie’ system is always faster or the same as walking alone. In the limiting case the rider doesn’t stop until the end point (there’s nothing in the OP’s description about dividing the course equally, rather it implies a set distance for each leg, so this is possible with a short enough course), so the initial walker ends up walking the whole way.
On a purely practical level, when two people walk together, they can only walk as fast as the slower walker. By eliminating the slower walker half the time (by his or her riding) it will speed up the over rate by that much.
Here’s another way to explain it without resorting to numbers.
Think of how this looks to just one walker. He walks for a while, then finds a bike and rides for while. He drops the bike and walks some more. Then he finds another bike (it’s actually the same bike) and rides some more. He does this until he reaches his destination. He never stopped, and rode part of the way and walked part of the way, so it must be faster than walking all the way. You can apply the same reasoning to the other walker. Now you have two people who traveled faster than if they both walked.
(I suppose it would be a little tricky to time it so they both reached the destination at the same time.)
Under the conditions imposed by Marilyn, I would agree with her (nobody ever stops and waits, and nobody walks the entire distance.)
Maybe it’s easier to imagine if we used a car instead of a bike.
Person A drives 1 mile at 60 mph (1 minute), gets out, and starts walking. Person B takes 10 minutes to walk a mile (they are both competitive speed walkers) and arrives at the car in 10 minutes, jumps in, and drives the next mile at 60 mph, passing Person A in the process. Person A, having walked for 9 minutes already, walks another minute to get to the car, jumps in, and drives another mile. At the end of 3 miles, person A walked for 10 minutes, and drove for 2 minutes (12 minutes). Person B, walked for 20 minutes, and drove for 1 minute (21 minutes.) A person walking the entire way needs to walk for 30 minutes.
At the end of 20 miles, Persons A and B would have driven for 10 minutes and walked for 100 minutes, or 110 minutes. A person who walked the entire way needs to walk for 200 minutes.
I would have said it’s obvious that walking + riding gets you there faster than walking alone. The only trick is how to explain away the “someone’s always walking, so that represents the true rate of progress” contention.
The actions of both persons are identical, and the distance covered is the same for each, so you can reduce the equation by thinking only in terms of one person. You can also ignore however many times they did the bike/walk swap, it is just a mulitplier of the basic action and isn’t relevant.
Each individual walked for half the time to cover the distance and rode for the other half of the time. That is faster than the individual walking the whole way.
Nevermind how the bike got where, that is a detail that isn’t required in order to make the math clear.
If (for example), the time to walk the entire distance is one hour, the total walking time would still be one hour. However, there are times when both people are walking at the same time. Hence the total time would be less.
I agree that this is where the mental catch lies. The catch is overcome when you realise that the true rate of progress is set by the rearmost person, and the rearmost person is not always walking. The true rate of progress is set by the person walking only when the person on the bicycle is in front of them. But for a non-zero part of the time, the person with the bicycle is behind, catching up to the person who had the bicycle earlier (but who is now walking). During this period, the true rate of progress is set by the person cycling.
