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.
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.