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escalator math - Alierak
June 14th, 2004
12:42 am

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escalator math
I'm developing a habit of walking up the up escalators at the Porter Square T stop. Not because I'm ever really in a hurry there, I guess, but because it pleases me to be able to do so without slowing down or getting seriously out of breath.

Today it occurred to me that there's an exercise tradeoff to be had. Do I want to go more slowly, in which case it's less strenuous exercise, or more quickly, in which case it's not a very long workout? I probably need the longer aerobic workout. But I at least feel like trying to quantify the tradeoff.

So, suppose the escalators rise at v_e feet per second, and are h_e feet high, with steps of height h_s (ignoring the shallow steps at top and bottom, and the flat mezzanine areas). If I take s steps per second, then I'll be moving at s * h_s feet per second relative to the escalator. That means I'll reach the top in h_e / (v_e + s * h_s) seconds.

I think h_e is about 100 feet, starting from the outbound platform and including all three escalators to the surface. I think v_e is about 0.5 feet per second, at least, it seems that slow, like it would be three minutes worth of standing around on escalators. I would guess h_s to be about 0.85 feet. So the tradeoff is something like

t = 100 / (0.5 + s * 0.85)

If I take one step every two seconds (s = 0.5), then my escalator climb takes t = 108 seconds, and I take a total of (t * s) = 54 steps. If I take one step per second (s = 1), then my escalator climb takes 74 seconds (and I end up taking 74 steps on the Porter Stairmaster). If I take two steps per second (s = 2), then my escalator climb takes 45 seconds, and I end up taking 90 steps.

Ok, that was silly. Going to sleep now.

Current Mood: sleepysleepy
Current Music: oscillating fan
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From:lite
Date:June 14th, 2004 11:43 am (UTC)
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*blink... blink... blink*

OK, but what about the very real possibility that one or more of your escalators, due to age, don't actually move at constant speed v_e, but actually a variable speed, v(t), where C_a v_e <= v(t) <= (1-C_b) v_e, where C_a and C_b are small constants (perhaps close to 0.05 or 0.1).
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From:alierak
Date:June 14th, 2004 12:50 pm (UTC)
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Well, I'm very sure they don't have a constant rate of vertical rise, hence "ignoring the shallow steps at top and bottom". This is already supposed to be an approximation.

But I sense you might be poking fun at me anyway as your inequality makes insufficient sense. It would mean something like 0.5 v_e <= v(t) <= 0.95 v_e, thus v(t) is always less than how fast I think the escalator should be going. Porter Square has the longest escalators in Boston, and typically one of them is down for maintenance every other day. So maybe that's not so inaccurate :)
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From:lite
Date:June 14th, 2004 02:51 pm (UTC)
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It is probable that I am mocking you gently.
I mock all my friends. I'm just a big dork like that :)
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From:kareila
Date:June 14th, 2004 01:12 pm (UTC)
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I can walk up the long escalator without straining myself, but if I walk up the second escalator to ground level as well, I start to get winded.

One day recently, all the long escalators were working and I walked all the way up, but the second level up escalator was broken and I had to walk the stairs. That was painful.
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