I'm taking the 'Introduction to mathematical thinking - Keith Devlin' and in the Problem Set Solutions, there's a question:
You've arrived late at an airport. You're rushing to catch your plane. Unfortunately, your gate is at the far end of the terminal. The fastest you can walk is a constant speed of four miles per hour. For part of the way, there is a moving walkway moving at two miles per hour. You decide to take the walkway and continue to walk. So you're going to walk all the way, but part of, when there's a walkway you're going to walk on the walkway so you can go faster, right? Just as you're about to step onto the walkway, you notice your shoelace is loose. And the last thing is you want to do is get your shoelace caught in the mechanism of the walkway. So you've got a choice. You either stop and fasten your shoelace just before you step on the walkway, or you step on the walkway and then stop to fasten your shoelace whilst the walkway is moving you along. And there's no other options. Those are the two options you have. Which option will get you to the gate fastest?
How do you determine which option to choose? You can't use distance-speed formula since you have neither been given the distance to the gate nor the distance of the walkway.
Think about yourself "splitting up" just before you enter that walkway. Instance 1 of you ties their shoelaces on the ground, then steps on the walkway. Instance 2 steps on the walkway, then ties their shoelaces.
After the (same) time it takes for both instances to tie their shoelaces, instance 1 is still at the start of the walkway, while instance 2 is farther ahead (as much as the 2mph walkway has transported them in the time). So at the same time they start again at their "full speed". Instance 1 enters the walkway and walks on it with 4mph in addition to the walkways own velocity of 2mph. Instance 2 does the same, adding its own 4mph walking speed to the walkway's 2mph.
It should be clear that instance 1 can never catch up with instance 2. At any point in space after the point where instance 2 started to walk itself on the runway, both instances move at the same speed. Since instance 1 arrives there later, it cannot catch up.
Sure, the distance in space will change: Once instance 2 steps off the walkway and has to continue alone on foot (4mph) the distance to instance 1 will shrink, which is still on the walkway and so is moving a 6mph relitive to the ground. But the same effect will happen the other way around once the next walkway appears: Instance 2 will get farther away from instance as long as instance 1 hasn't stepped on the walkway itself.
The 2 instances aren't seperated by the same distance, but they are separated by the same time. If you could measure the time from when instance 2 arrives at a point fixed to the ground until instance 1 arrives there, you'd always get the same time difference, no matter where you measure.
How long is that time? Well, it's the same at any point, so let's measure it at the point where instance 2 is when it is done with the shoelaces and starts to walk in the walkway.
As I said above, this will be at a distance from the start of the walkway that corresponds to the walkway's 2mph times the shoelace tieing time. Instance 1 however approaches that point with 6mph, so will only need a third of the time it uses to ties it's shoelaces.
So that's the final result: If you tie your shoelaces on the walkway, you will arrive earlier than if you do it on the ground, and the amount of time your are earlier is a third of the time it takes to tie your shoelaces.