A Swedometer First

Welcome to the first Swedometer blog post. As hinted in the FYI section, topics will revolve around math and physics modeling. To kick off we take a look at the relative motion of walking up and down an escalator. What is faster, completing the round walk on a moving or a stationary escalator?

I first saw this problem back in my early teens. A friend had taken part in a mathematics contest where it featured in some form or another. I remember another friend and I was quite intrigued at the time, as we thought it was quite a challenging problem.

Solving a problem requires understanding the problem. Well, at least developing an understanding while solving it. It is also important to choose a suitable set of conceptual “tools” for the job. These things typically get easier with experience.

The escalator problem is actually quite easy to solve with a bit of algebra, and only the four basic arithmetic operations are needed. But we will try to also argue the solution through visualization, for better understanding.

Understanding And Modelling The Problem

So what do we need to understand about this problem? Well, let’s start with the fact that velocity is always relative to something. The walker is moving relative to the escalator, but also relative to the surrounding. We could also argue that the escalator, or the surrounding, moves relative to the other two. This discussion of relative motion concerns the choice of reference frame, an important part of building a physics model.

Often one can find a reference frame that is “easier” to work with than others. In this case, it would be easy to solve the problem in any of the reference frames mentioned, but we will choose the surrounding as reference. This is probably the most natural for most people, as we tend to think of our surrounding as a fixed background in some sense.

In order to build the model we need to make some assumptions about the system (parts and interactions). First we assume that both walker and escalator move at steady pace. A bystander sees the walker making his way to the top of the escalator, turn around and walk back down. Further, we assume that the walker speed relative to the escalator is the same both ascending and descending.

The speed of the walker in the surrounding reference frame is a combination of the walker and escalator speeds. If we assume that the escalator is moving up (direction does not actually matter for the final answer) we can express the up and down speeds of the walker as seen by the bystander

$$v_{up}=v_{walk}+v_{esc} \tag{1}$$
$$v_{down}=v_{walk}-v_{esc} \tag{2}$$

Let the escalator length be $$L$$. The total walk time is then

$$t = \frac{L}{v_{up}} + \frac{L}{v_{down}} \tag{3}$$

Next we take a closer look at these equations.

Taking The Problem To The Limit

To better understand a problem it is often useful to take it to extremes. And no, not going crazy, but finding limiting cases where the problem kind of breaks down. For example, we may look at the relation between walker and escalator speeds.

What is the slowest escalator speed? Zero, a stationary escalator. What is the fastest escalator speed? Well, if the escalator moves faster than the walker, the walker will be unable to walk against the escalator. This means we need not consider escalator speeds faster than the walker speed. So the upper limit is an escalator speed equal to the walker speed, in which case the walker will never finish. In mathematical terminology, we say that the time taken goes to $$\infty$$ (infinity).

So to recap, we have a finite time for a stationary escalator, and an infinite time for equal walker and escalator speeds. All other escalator speeds of interest lie in between these two limits. Should not all travel times of interest, in consequence, lie in between those two limiting times? If so, we have already found our answer, the stationary escalator would result in the fastest walk.

It is probably possible to prove this mathematically using theorems of monotonic functions. But we will leave that for undergraduate calculus classes, and instead focus on visualizing the relative motion.

Graphical Results

The video below shows a comparison between walking up and down a stationary and a moving escalator. In this example the walking speed is twice that of the moving escalator. The result is a clear victory for the walker on the stationary escalator.

This playlist holds more examples with different walking speeds. All of them show the walker on the moving escalator taking an early lead, only to be overtaken on the way down by the walker on the stationary steps.

To further convince ourselves that a stationary escalator is indeed quicker to walk up and down, we aim for a more general representation. The answer should be independent of the length of the escalator, so let’s try to find an alternative expression to equation (3), which is independent of $$L$$. To do this we express the time taken for a moving escalator relative to the time taken for a stationary escalator, as $$t_{rel}$$. And we express the escalator speed relative to the walker speed, as $$u=\frac{v_{esc}}{v_{walk}}$$. Using equation (3) we can then express the relative time as

$$t_{rel} = \frac{1}{1-u^2} \tag{4}$$

The graph below plots equation (4). The reference is the stationary escalator, with $$u=0$$, which takes a relative time $$t_{rel}=1$$. It is then seen that a moving escalator, with $$u>0$$ always takes longer time $$(t_{rel}>1)$$. Hence, it follows that walking up and down a stationary escalator is the faster alternative.