A problem in mechanics This is an apparently simple problem that I could not solve. The source is a colleague who interviewed for the European space agency.
You are standing in a field with a chronometer and a camera. Someone throws a ball (a point mass) in the air, with an unknown speed and velocity. You can take pictures with the camera and record the time between pictures with the chronometer, but you can’t directly measure the distance between you and the ball. How many pictures do you need to take to determine where the ball will hit the ground?
Assume that the earth is flat, that $g$ is known to arbitrary precision, that air has no effect and that the trajectory is generic. Assume moreover that the camera is at a known position and at a known height above the ground (may assume that the height is $0$ if useful for the solution).
Attempts at solution
The state of the ball is determined by six parameters, but we only want to know where the ball hits the ground, so we may need only five or less parameters.
The first picture gives two parameters and the second one three, so two pictures may be sufficient. My gut feeling is that they are not, but I wouldn’t know how to prove it.
The third picture gives three new parameters, but they cannot all be independent. Indeed at most one is, so the answer should be three, but again I could not find a way to prove it.
I refuse to assume the earth is flat...
But I would guess four, because we are talking about a parabolic curve, which is represented by y = ax²+bx+c. We need to determine the three coefficients a, b and c, but we only obtain the information of the difference between two photos. So to get three differences we need four points.
I think the answer would be that the problem is unsolvable. No matter how many pictures you take, range cannot be predicted. Let me illustrate with a more intuitive situation. Suppose you live in a high-rise building, and you have one wall made of glass. You cannot look down on the ground, and don't know how high up you are. You can only look straight ahead. Now, someone from the ground throws a ball up in front of you. You can see the ball go up from a certain point on the floor of your room, reach a maximum height, and go down again from some other point. Suppose you start taking pictures using the same set-up you had before. Would you ever be able to know where the ball will hit the ground? Of course, you can assign origin to be that point where the ball first comes into your sight, have an $x$-axis along the floor of the room and a vertical $y$-axis, and let the time be $t=0$ at that instant. If you then take one other picture, that would be enough for you to draw the trajectory of the ball with respect to the coordinate system I described. However, since you do not know what the equation of the horizontal &quot;ground line&quot; is, that is, don't know how far below you the ground lies, you can never figure out where exactly the point of projection of the ball was, or where it will end up.
Hope this makes it clear.