Some teenagers are exploring the outer perimeter of a castle. They notice a spy hole in its wall, across the moat a horizontal distance x and vertically up the wall a distance y. They decide to throw pebbles at the hole. One girl wants to use physics to throw with the minimum speed necessary to hit the hole. What is the required launch speed v and launch angle above the horizontal? The situation is sketched in Fig. 1.
The trajectory of a stone launched with optimal speed v and angle θ such that it passes through a hole located at rectangular coordinates (x, y) relative to the launch point. In polar coordinates, the hole is located at (r, ϕ).
The trajectory of a stone launched with optimal speed v and angle θ such that it passes through a hole located at rectangular coordinates (x, y) relative to the launch point. In polar coordinates, the hole is located at (r, ϕ).
An alternative version of this problem is2,
Find the maximum r and corresponding θ for given v and ϕ,
Another related problem is
Find the envelope of the projectile motion in the uniform gravitational field for the fixed initial speed v of the projectile.