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Math Help - Expected number of bounces

  1. #1
    Senior Member TriKri's Avatar
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    Expected number of bounces

    Say we have the following scenario:

    A ball is bouncing forth and back between two parallel walls, located at a distance from each other so that the ball (which is always moving parallel to the normal of the walls) will have to travel one meter to get from one wall to the other. In each bounce the ball will get a new speed by the wall it is bouncing on, which is uniformly distributed between 0 and 1 meter per second. The velocity is of course directed against the other wall from where the ball is located.

    At time t_0 the ball bounces against one of the walls. What is the expected number of bounces the ball will make after t_0 if the ball is stopped after ten seconds?
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  2. #2
    Moo
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    Let N be the number of bounces.

    Then N is such that X_1+\dots+X_{N-1}<10 and X_1+\dots+X_N>10, where X_i denotes the speed at which the ball travelled before the i-th bounce.
    (In particular, X_i are iid and following a uniform distribution over (0,1))

    Does it help ? I must say I don't have much time, and haven't finished the problem yet, so I hope you can do some things with that.
    Last edited by Moo; November 30th 2009 at 02:29 PM.
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    Senior Member TriKri's Avatar
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    Quote Originally Posted by Moo View Post
    ... where X_i denotes the distance (or the speed, they're the same number) the ball travelled.
    No they are not the same, the distance it travels is always 1 [m], while the speed is uniformly distributed between 0 and 1 [m/s].
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    Moo
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    Sorry, I edited. I hope it's clearer...
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    Senior Member TriKri's Avatar
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    I have made some work with this problem, and this is what I have found.

    We will first define the function E(t), as the expected number of bounces after t seconds. Of course, E(t) = 0,\text{ if }t \leq 1, since 1 second is the least amount of time the ball can take to travel from one wall to the other. But what about if t > 1?

    Suppose that we call the speed of the ball s. The time \Delta t it takes for the ball to travel between the two walls is 1/s. If \Delta t < t, then the ball will reach the other wall and bounce on it. \Delta t < t in turn yields that s > 1/t. If s < 1/t, the ball will not bounce even once. But if s is between 1/t and 1, it will bounce, and the expected number of bounces after that bounce will be E(t-\Delta t) (since t-\Delta t is the time the ball has left), making the expected total number bounces 1+E(t-\Delta t)=1+E(t-1/s).

    Let's define the functin E(t, s) to be the expected number of bounces after t seconds, if we know that the initial speed will be s. We then have:

    E(t,s)=\left\{\begin{array}{ll}<br />
0,&\text{if }t<1\text{ or }s < 1/t\\<br />
1+E(t-1/s),&\text{otherwise}<br />
\end{array}\right.

    Also, since s is uniformly distributed between 0 and 1, we can conclude that E(t)=\int_0^1 E(t,s)ds, wich gives (by substitution):

    E(t)=\left\{\begin{array}{ll}<br />
0,&\text{if }t<1\\ \\<br />
\displaystyle{\int_{1/t}^1 (1+E(t-1/s))ds},&\text{otherwise}<br />
\end{array}\right.

    which can be rewritten as

    E(t)=\left\{\begin{array}{ll}<br />
0,&\text{if }t<1\\ \\<br />
\displaystyle{1-\frac{1}{t}+\int_{1/t}^1 E(t-1/s)ds},&\text{otherwise}<br />
\end{array}\right.

    This is not a very nice function. Although E will be continuous, it's derivative won't be continuous since after t=1. It's second derivative won't be continuous since after t=2, etc. The best shot is probably to approximate the function numerically. It may also be possible to solve it separatelly for n < t \leq n+1, and iterate n over the natural numbers. I doubt it works very well for large n though, since each expression builds on all n-1 previous expressions for E, and will probably look very bad just after a few iterations.
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  6. #6
    Senior Member TriKri's Avatar
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    Quote Originally Posted by Moo View Post
    Sorry, I edited. I hope it's clearer...
    One of the first things you should check when you have put up an equation and you are not really sure whether it holds or not, is if the two sides have matching units. In your case, X_i represents a speed [m/s] while the number 10 represents a time interval [s], you therefore have a unit mismatch and you can't compare the two sides.
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