Free-Throw Entry-Angle Lab

How the entry angle opens or closes the hoop, and which launch angle scores most when your release is not perfect.

Presets

Shooter

Distance to hoop

Side angle

Release height

Shot

Launch angle

Backspin

Aim target

Board spot, side

Board spot, height

Perfect-aim speed solver (finds the speed for this angle)

Release speed

Depth error

Add human release scatter to each shot

Scatter amount

Depth error moves the crossing point along the flight direction (+ = long). Scatter (1 sigma, x amount): 1.2 % speed, 0.6 deg angle, 0.45 deg aim, 0.25 rev/s spin.

World

Environment

View

Trajectory preview Force vectors Ghost of last shot High quality

Slow motion

Live readouts

Side view: trajectory and entry angle

Looking down the flight path

The rim seen along the flight direction is an ellipse with semi-axes R and R sin(alpha). The ball (radius r) only gets through untouched if its centre stays inside the dashed green region.

Experiments

Pick an experiment. They use the current distance, release height, backspin and environment.

The science

Flight. m dv/dt = m g - ½ρA C_d|v|v + ½ρA C_L|v|² (ω̂ × v̂), RK4 at 2 ms. Ball 0.62 kg, r = 0.1195 m, C_d = 0.5, C_L = 1/(2 + v/(rω)).

Geometry. Rim inner radius R_in = 0.2286 m, wire radius 8 mm, centreline radius R_c = 0.2366 m, height 3.048 m. Free-throw line 4.191 m from the hoop centre.

Entry angle and the clear window. Seen along the flight direction the rim is an ellipse with minor semi-axis R_c sinα. The ball only passes without touching if the front and back rim points stay at least r + ρ from its path: W(α) = 2 (R_c - (r + ρ)/sinα). It closes completely at α_min = asin((r + ρ)/R_c) = 32.6 deg and widens to 21.8 cm at 90 deg.

Minimum speed. Without air, the slowest shot to a point (s, Δy) leaves at tanθ = (Δy + √(Δy² + s²))/s with v² = g(Δy + √(Δy² + s²)). Its entry angle is only about 39 deg for a free throw, so a slightly higher arc trades extra speed for a wider window.

Rim and board contact. Sphere against torus (closest point on the ring), impulsive normal restitution 0.62 and Coulomb friction 0.35 with spin coupling (thin-shell inertia 2/3 m r²); board restitution 0.66. Backspin makes a rim bounce lose forward speed, which is why soft-looking shots drop.

Assumptions. Rigid sphere, rigid rim and board, no net force except a simple damping and funnelling zone, no wind, spin axis horizontal and perpendicular to the shot, a shot counts as made when the ball centre passes inside the rim circle and then 30 cm below it.