the excercise is the following:
a man throws a non-rotating ball from a height 2m into a 3m high basketball hoop. The center of the hoop, which has a diameter of 0.5m, is 5 m in front of the man and the ball has a diameter of 0.25m.
Now the question is: If the man throws the ball directly(that means without touching anything like the plate behind the hoop or the frame of it) into the basketball hoop, in which directions and between which velocities can he choose to throw the ball to do so?
and there is a further approximation, that the velocity of the ball, when falling through the ring can be considered as constant.
well the equations should not be that hard:
when you choose connecting line between the man and the basketball hoop as the x-axis, then we have
further, we have some geometric restrictions, but they are fairly subtle, like e.g. that the ball has to surpass the frame of the hoop and that he does not touch anything of it, when falling through the hoop, but all of them somehow depend on the point where the ball falls through the ring, which raises a problem.
The Attempt at a Solution
well, my problem is, that i tried to solve this problem, without using the approximation of constant velocity, because I do not know how this could help and failed, more or less because the equations became far too cumbersome. I am just looking for a promising approach.