How many steps on an escalator?

One of the joys (or burdens?) of being fascinated by mathematics is that you see interesting mathematics problems all around you in your everyday life!

I have recently acquired one of those little devices you attach to your belt that counts the number of steps you make in a day. The result of this is that you get obsessional about maximising your number of steps in every situation. There is even a temptation to mince along with very short steps rather than my usual manly stride, just to increase the step-count!

Today, I had to go up an escalator in a shopping centre. I never stand still on an escalator, unless it is totally blocked with other people. So, I found myself thinking about how many steps might I manage going up this escalator. It seemed intuitive to me that the faster I went the more steps I would get in before I reached the top. After all, if my speed was zero (i.e. standing still on one step) then my step-count for the escalator ride would also be zero.

Intuitively I imagined that there would be a linear relationship here. So, if I went twice as fast I would get in twice as many steps before I reached the top. But, wait a minute! That can't be true. Because I could never get more than the total number of visible steps on the escalator, however fast I went. In fact, to get in a number of steps equal to the number of visible steps on the escalator I would have to travel at an infinite speed!

This was getting so interesting, I forgot to do my shopping and just walked home puzzling it out.

It's a really nice mathematical problem. I managed to calculate, for example, that ...

• if there are 30 visible steps on the escalator (i.e. the number of steps you would see if the escalator were stationary), and
• the steps are disappearing at the top at the rate of 90 steps per minute, and
• I go up the escalator at the rate of 120 steps per minute (pretty fast),

then I will add 17 steps to my step-counter.

If I went twice as fast up this escalator, doing 240 steps per minute, I calculated that I would increase my step-count to 21 or possibly 22 steps. And at half the speed, i.e. 60 steps per minute, I would do 12 steps before reaching the top of the escalator.

I'll leave this as a challenge for readers ... can you produce a formula for the number of steps I add to my step-counter (m) if there are p visible steps; the steps are disappearing at the top at the rate of s steps per minute; and I go up at the rate of n steps per minute?

m  = ....?

And what would the graph look like, if you plotted m against n?

Solution in my next post.