Cannon ball sweep
A cannon placed on the ground shoots cannon balls at any angle between 0° (a horizontal shot which is assumed to reach zero distance) and 90° (a vertical shot, not recommended) with a fixed initial speed. All orbits are in the same vertical plane and a shot finishes when the cannon ball hits ground level.
If the maximum horizontal distance it can shoot is 300 meters, what is the area in square meters of the set of all points in the vertical plane that can be reached by a projectile.
Assume that both the cannon and projectiles are of negligible size and air resistance can be ignored.
The answer is 30000.
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1 solution
From classical mechanics it follows that the projectile orbits are parabola. It is not difficult to show that the maximum horizontal distance (=range), is obtained by a 45° shot and equals the square of the initial velocity divided by the gravitational acceleration g = 9,8 m/s^2. On the other hand, the maximum height is obtained by a vertical shot and equals half the horizontal range.
To determine the limiting curve of all points that can be reached in the vertical plane one can proceed as follows: write the equation for the vertical coordinate (say y) in terms of the horizontal coordinate (say x) containing the initial shooting angle (say alpha) as a parameter Of course this is the equation for the parabolic orbits of the projectiles. Now fix x and maximize the y coordinate as a function of alpha. These maxima y as a function of x form the limiting curve which turns out to be a parabola with vertex at the maximum vertical height ymax (see above) and passes through the point of maximum horizontal distance xmax (see above).
To calculate the area under this limiting curve (=parabola) either integrate from x= 0 to x = xmax, or note that it is equal to 2/3 of the area of the bounding box [ (0,0),(xmax,0),(xmax, ymax),(0,ymax) ] and since ymax = 1/2 xmax the area is equal to 1/3 times the square of the maximum range.
For a maximum range of 300 meters this gives an area of 30000 square meters.