(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that on the time interval [0,h] we have a force acting along the x-axis with magnitude [itex] 0 \leq f(t) [/itex] at time t, and f(0) = f(h) = 0. For a particle of mass m starting at rest, so that its distance x(t) from 0 satisfies 0 = x(0) = x'(0), show that [itex]x'(h) = \int_0^h f(t) dt[/itex].

3. The attempt at a solution

This is one of those problems where I swear I am right and the book is wrong.

Let P denote the particle. Then on the interval [0,h], the forcef(t) acts on P along the x-axis so henceforth I will writef(t) = (f(t),0) = f(t). The second law tells me that mx'' = f(t). Hence

[tex]\int_0^h f(t) dt = \int_0^h m x'' dt = \int_0^h mv' dt = mv(h) - mv(0) = mv(h)[/tex]

since v(0) = x'(0) = 0. So this seems to contradict what I was suppose to show except in the special case that m=1. I haven't really used my assumption that f(0) = 0 = f(h), but I can't think of how that would tie into this problem other than tell me that there is no net force acting on P at both t=0 and t=h. Would someone mind giving me some pointers on this one?

- Sam

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# Homework Help: Impulsive force question

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