Optimizing a Free Throw: Solving for Initial Speed and Angle

[Metaphysiks]

Homework Statement

Goal: Find an equation for making a free throw.
Directions:
Write an equation for v in terms of g, x, y, theta_i, z eliminating t. Then separately write z in terms of r, R, and theta_f and also write theta_f in terms of y, x, and theta_i.

Variables:
v = initial speed
theta_i = initial angle
x = range
y = elevation
theta_f = final angle
r = radius of ball
R = radius of rim
z = variation in x (allowable for swish)

Homework Equations

Displacement:
x=(vcos(theta))t
y=(vsin(theta))t-(gt^2)/2

Final Velocity:
Vx=vcos(theta)
Vy=vsin(theta)-gt

The Attempt at a Solution

 

[berkeman]

Homework Statement

Goal: Find an equation for making a free throw.
Directions:
Write an equation for v in terms of g, x, y, theta_i, z eliminating t. Then separately write z in terms of r, R, and theta_f and also write theta_f in terms of y, x, and theta_i.

Variables:
v = initial speed
theta_i = initial angle
x = range
y = elevation
theta_f = final angle
r = radius of ball
R = radius of rim
z = variation in x (allowable for swish)

Homework Equations

Displacement:
x=(vcos(theta))t
y=(vsin(theta))t-(gt^2)/2

Final Velocity:
Vx=vcos(theta)
Vy=vsin(theta)-gt

The Attempt at a Solution

That's a good start, but you should factor in the height that the ball is released from, and the final height as it passes through the hoop. Drawing a sketch should help you.