- #1
Gil-galad
- 11
- 0
Hello! Since this is my first post I would like to say I am happy that I joined PF and I will try to contribute as much as I can to this great community.
So let's get down to the gist of the problem.
You are standing 4m from a vertical wall and 0.8m from the ground. You have a bouncy ball that you throw with 10m/s initial velocity at some angle 5^{0} \leq \alpha \leq 45^{0} from the horizontal towards the ground. The ball bounces from the ground then bounces off the wall losing 10% of its initial velocity after each bounce. The question is to find this initial angle \; \alphawith less than 0.01^0 of an error, so that the ball returns to the thrower (original position) assuming that the ball bounces off the wall/ground with the same angle as it hits it (incident angle=reflected angle). Assume gravity at 9.81 m/s/s and no friction is involved. The ball does not rotate.
So my question is how do you approach such problem? Is it possible to construct an analytical expression and then perhaps maximize/minimize only with the help of trivial kinematic equations? Or one must use some generalized coordinates and langrangian & hamiltonian mechanics? I would appreciate some guidance.
I pondered quite for some time before my calculations became quite complicated and nearly gave up. I used the trivial kinematic equations for the trajectory of the ball and tried to get an expression involving the initial angle related to the subsequent reflected ones that define the other 2 trajectories but to no avail.
So let's get down to the gist of the problem.
You are standing 4m from a vertical wall and 0.8m from the ground. You have a bouncy ball that you throw with 10m/s initial velocity at some angle 5^{0} \leq \alpha \leq 45^{0} from the horizontal towards the ground. The ball bounces from the ground then bounces off the wall losing 10% of its initial velocity after each bounce. The question is to find this initial angle \; \alphawith less than 0.01^0 of an error, so that the ball returns to the thrower (original position) assuming that the ball bounces off the wall/ground with the same angle as it hits it (incident angle=reflected angle). Assume gravity at 9.81 m/s/s and no friction is involved. The ball does not rotate.
So my question is how do you approach such problem? Is it possible to construct an analytical expression and then perhaps maximize/minimize only with the help of trivial kinematic equations? Or one must use some generalized coordinates and langrangian & hamiltonian mechanics? I would appreciate some guidance.
I pondered quite for some time before my calculations became quite complicated and nearly gave up. I used the trivial kinematic equations for the trajectory of the ball and tried to get an expression involving the initial angle related to the subsequent reflected ones that define the other 2 trajectories but to no avail.
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