Physics Challenge Problem

If you can get this problem, you have mastered 2D kinematics

1. A basketball player who is standing 15 feet away from a basketball hoop is trying to make a basket. If the height of the hoop is 10 feet, and the height at which the player shoots the ball is 6 feet, at what angle and with what speed should the player shoot the ball?

 Pre-Step1: Define all known variables

 

STEP 1: Find the amount of time required:

 This gives me two possible solutions for time

Note that these two times will be the exact same time if the root is zero. IF:

However, if the root is rational and greater than zero, t_2­ will be the shorter time. This is the point when the ball is moving upward past the hoop, and therefore cannot be the time when the ball sinks into the basket from above. Therefore we will use t₁ for the rest of the solution.

Note that, for “t” to be a real number, v_iy² must be larger than or equal to 4(1/2)(a)(Δy)
this will keep the root rational.. You can actually use this reasoning to see that any initial velocity in the “y” direction greater than or equal to 4.888386 will work for v_iy

STEP 2: Find v_i in the “x” direction:

In order to solve for v_ix you must pick a value for v_iy that is ≥4.888386

 -Note that the term v_iy is only in the denominator, so the larger value for v_iy the smaller v_ix  will be.

Since a true answer will be obtained from any value of velocity in the y direction that is greater than or equal to 4.888386, we can just choose our initial y-velocity. For simplicity, we will stick with 4.888386 

STEP 3: Use Pythagorean theorem to solve for v_i­

STEP 4: Use Sin property to find the angle