Interesting question. Note that we are considering a Classical trajectory in this problem, which means that we know exactly what the different coordinate positions are. Specifically, let the positive z direction be the direction from the gun to the pinpoint target. Using a typical (that is to say left-handed) coordinate system we may define the positive y direction to be up and then the x direction sticks out horizontally from the yz plane. We know without error that the x component of momentum is zero (barring wind, etc.) This requires that the x component of displacement is completely unpredictable. Thus we have absolutely no ability to predict if the bullet hits the target.Originally Posted by hongster5
Ignoring this point we can focus on the y components of momentum and displacement. I am assuming the gun and target are at the same height for the sake of simplicity, so we can use Classical Physics to predict the angle the gun has to be at to hit the target. What will happen is that the magnitude of the y component of the final momentum (when it hits the target) will be the same as the magnitude of the y component of the initial momentum. The problem is, of course, that we know the magnitude of the momentum, not the root-mean-square value of the error (more or less simply called the "error") in the momentum.
There are various and sundry ways to estimate an error in the momentum. One argument (which I like because it tremendously simplifies things) is to assume a momentum measurement will take not a Gaussian distribution, but a Poisson distribution, in which case the error in the measurement will simply be . (This is justified by considering the total momentum as the sum of a large quanta of momentum.) I honestly don't know the range of validity of this assumption, but we have to have some way to calculate an error for this problem. You may wish to check with your professor about how to get an error for the momentum.
Assuming the error estimate above is valid, we can now calculate an error for the displacement in the y direction.
This number should be fairly small, probably undetectable, as makes sense for the result of the uncertainty in a Classical measurement, so the bullet will be fairly close to the target as far as the y direction is concerned.
We can do a similar analysis using the x component of the final momentum.
(Of course, these results are completely overshadowed by the result in the x direction!)