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What is the average of fall distances

  1. Mar 28, 2015 #1
    [no template, as this post was moved from here from the Quantum Mechanics forum]

    In griffiths 2nd quantum mechnics,

    problem : Suppose I drop a rock off a cliff of height h. As if falls, I snap a million photographs, at random intervals. On each picture I measure the distance the rock has fallen.

    Question : What is the average of all these distances? That is to say, what is the time average of the distance traveled?

    I know p(x) is 1/2(hX)^(1/2) and can solve the problem like : Integral(from 0 to h)x dx/2(hX)^(1/2)

    other method is : Integral(from 0 to T) x dt/T

    but I don't know how it's possible. that means p(t) = 1/T .. but T is constant{ (2h/g)^1/2}

    and I don't know meaning of dt/T
     
    Last edited by a moderator: Mar 28, 2015
  2. jcsd
  3. Mar 28, 2015 #2

    HallsofIvy

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    Here's how I would do that- the constant acceleration due to gravity is -g. The velocity, at time t isv(t)= -gt+ v0. Since the rock was "dropped", rather than thrown up or down, v0= 0 and v(t)= -gt. Then the distance traveled in t seconds is [itex]s(t)= -(g/2)t^2+ h[/itex]. The rock will hit the bottom when [itex]s(t)= -(g/2)t^2+ h= 0[/itex] so when [itex]t= \sqrt{2h/g}[/itex]. The average of a function, f(t), for t between a and b, is [itex]\int f(t)dt/(b- a)[/itex] so the average height, for t between 0 and [itex]\sqrt{2h/g}[/itex] is [tex]\frac{\int_0^{\sqrt{2h/g}}( -(g/2)t^2+ h) dt}{\sqrt{2h/g}}[/tex]
     
  4. Mar 28, 2015 #3

    mfb

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    I don't understand your notation - I think you mix different things here. p(t) in the first approach looks like the position, while in the second approach it seems to be a probability to take a picture at a specific time. But then the integral should have the position as function of time, not x.
     
  5. Sep 2, 2015 #4
    I think the problem and his approach was worded so strangely. I had to just close the book and think about it by myself, and figure out how I'd define each piece of the problem mathematically. Then how I could manipulate each piece in order to integrate it. Turned out it was pretty much identical.

    Sorry I necro'd this thread. I just had to see if anyone else struggled with this example. Then I wanted to say how I approached the problem so anyone else in the future can maybe try the same type of thing if they were also confused.
     
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