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ladders
a lot of us use ladders can you sort this out?
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A Ladder Will Fall Infinitely Fast when Pulled
Consider a ladder of length L leaning against a frictionless wall which is at right angles to the ground. You pull the bottom of the ladder horizontally away from the wall, at constant speed v. The claim is that this causes the top of the ladder to fall infinitely fast.
Common sense tells us this can’t possibly be true, but can you find the flaw in the following supposed “proof” of this claim?Wall
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y| \
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Step 1: As shown, let x denote the horizontal distance from the bottom of the ladder to the wall, at time t.
Step 2: As shown, let y denote the height of the top of the ladder from the ground, at time t.
Step 3: Since the ladder, the ground, and the wall form a right triangle, x^2 + y^2 = L^2.
Step 4: Therefore, y = sqrt(L^2 – x^2).
Step 5: Differentiating, and letting x’ and y’ (respectively) denote the derivatives of x and y with respect to t, we get that
y’ = – (x x’) / sqrt(L^2 – x^2)
.Step 6: Since the bottom of the ladder is being pulled with constant speed v, we have x’ = v, and therefore
y’ = – (x v) / sqrt(L^2 – x^2).
Step 7: As x approaches L, the numerator in this expression for y’ approaches -Lv which is nonzero, while the denominator approaches zero.Step 8: Therefore, y’ approaches – infinity as x approaches L. In other words, the top of the ladder is falling infinitely fast by the time the bottom has been pulled a distance L away from the wall.
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7 Replies
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Firstly, I don’t understand a bleeding word of all that mathematical stuff so I didn’t even bother reading it. Secondly, ‘infinitely fast’ What does that mean? If it is supposed to mean the speed of light, then no. All rubbish. Speed of light = infinite mass = infinite energy required.
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i used a ladder once while drilling. the ladder wasnt at full stretch but was high. i consider myself pretty safety concious, but still do have the odd cut here and there :lol1: anyway..
up the ladder i was. “indoors ona concrete floor” not hammering not wiggling etc. i suddenly heard a bang and saw a white flash!
next thing i knew i was stairing at the ceiling and a few of our guys had run up to me… what had happend?the bang was the ladders hitting the floor.
the white flash i saw was the result of the back of my head hitting the concrete floor. even though my entire body was lying ontop of the ladders like a stretcher.
i still had the power drill in my hand! but with only half a drill bit? the other half was still in the wall??? it snaped!
one of the lower rungs on the ladder was bent were my foot hit while landing… (these were not crap ladder but industrial alluminium made ones.im not scared working at hights if i am safe enough, i am always of the opinion if im about to fall or something ill jump clear or land ok or something? how wrong was i?
one minute i was drilling away, next i was flat on my back without any warning what so ever… scared the macaroni cheese out of me! 😮that accident and the one i had with a drill & my finger taught me valuable lessons…. 😉
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Should’nt this experiment be conducted in a complete vacuum to negate the effect of air resistance slowing down the ladders fall 😮
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I did a good one once. I unknowingly had stood the ladders on a manhole cover under some gravel. When they inevitably slipped, my legs went through the rungs just as the ladders went through a window and came to rest on the ledge with my feet about 1 inch off the ground. I didn’t do that again!
I didn’t have time to check the velocity of the ladder as it fell, but I’m pretty sure it was a long way short of the speed of light. But then, of course, travelling at that speed, the light from my watch would never have reached my eyes as I was approaching lightspeed.
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Much the same as Rob,
In a school gym with a concrete and “varnished” floor. I am happy at height was working up at around 4m just drilled and screwed in a panel on to the wall lent out to push down the white screw cap. Felt a shudder filled my pants the ladder slid then stopped, for a splet second thought i was safe, oh no i wasn’t. Rode the ladder all the way to the floor bruised up whole left hand side of my body bloody big swollen ankle.
Not done that job again but will be invested in an anti slip mat an those circular feet that you can get should a job like that come up again.
“infinite fast” what about terminal velocity?
More importantly what about the ground?
Not infinite but definite!!!!
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Contrary to popular belief – falling doesn’t hurt! Stopping on the other hand 😉
Mark
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Theres one flaw and the ladder is standing on it!
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