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First to five. The physics is how you win.
- Skill this builds
- Predict where a thrown ball lands from its release angle and speed, calculate the best angle for a target above your hands, and read your own shooting percentage as the small sample it is.
- About
- About 10 minutes · Standards: NGSS, Outpost Education Library · Plays offline
Set the angle and the speed, then shoot. First to five points takes it. Arrow keys work too — up and down for angle, left and right for speed, space to shoot. Watch where their shots go in from; they are not guessing.
The physics is real: projectile motion integrated at a fixed timestep with g = 9.80665 m/s². The ball is treated as a point with no air drag and no spin, which is a simplification stated rather than hidden — a real ball loses a little to drag and gains a little off the rim, and both are small at these speeds. The steadying effect of holding your aim models a real research finding about gaze before movement; it is a model, not a measurement of you. Nothing here uses a camera and nothing about how you played leaves your device.
Playing as
Your Beacon
First to five. Take your shot.
You
0
—
Them
0
You stand
6.0 m
First to five
This run: Their cold night · They came to play
What can differ between games
Two of these are drawn each game, and the scoreboard names them. They bend the opponent's scatter and how the quiet-eye model behaves — never gravity, the rim, or your own body settings.
- They came to play — Their arc barely varies tonight. Bad news for the scoreboard, good news for you: a consistent shooter is the answer to the angle problem, drawn over and over, right in front of you.
- Their cold night — Their shots spray wide of their own habit. More room for you to take the game — and a scattered arc is much harder to learn anything from.
- A loud gym — Harder to get quiet in here. Settling your aim takes longer, and a rushed release drifts further than usual — the gaze work is most of the game tonight.
The physics
The rim is at 3.05 m. You release at 1.95 m, so the target sits 1.10 m above your hands — and you stand 6.0 m out, which is the third number in the problem.
Forty-five degrees is the best angle only when the target is level with your release. It is not, so the minimum-speed angle from where you stand is 50.2° at 8.4 m/s — computed live from your own release height and your own spot, so walk, or change the leg drive, and watch the answer move.
θ = 45° + ½·arctan(Δh ÷ d), with g = 9.80665 m/s². Point ball, no air drag and no spin — a real ball loses a little to drag and gains a little off the rim. Both are small at these speeds. Stated rather than hidden.
The body
Force in a shot starts at the floor and travels up — legs, trunk, shoulder, elbow, wrist. That is the kinetic chain, and it is why the arm-only setting caps out at 8.2 m/s while the leg drive reaches 10.6.
The legs also raise your release by 0.40 m, which shrinks Δh and lowers the angle you need. Two effects from one movement, which is why coaching starts at the feet.
The rings painted on the gym floor are where each body runs out: past 5.65 m the arm alone cannot reach the rim at any angle, and past 10.73 m neither can the legs. Both are computed from the caps, not painted by hand — walk out past the first ring with the arm only and feel the floor prove it.
Throwing, catching, jumping and balancing are taught, not grown — they do not reliably arrive with age alone.
Your own numbers
Take some shots and this fills in. It is a number about you, generated by you, compared to nobody.
The eyes
Adjusting your aim and pausing before you shoot tightens the shot here. That models a real finding — expert performers fixate a target for longer before starting to move, and training that gaze has improved accuracy in controlled studies.
Honest about what this is: the steadying you feel is a model of that research, not a measurement of your eyes. Nothing here uses a camera, and nothing about how you played leaves this device.
Here is what the game was doing while you played it. Forty-five degrees is the best angle only when the target is level with your hands, and a hoop never is — which is why the answer sits near fifty-one, and why it moves the moment you turn on the leg drive. That is the opponent’s whole trick. The rest of the shot is the same story in other subjects: the force starts at the floor rather than the arm, the arc is a parabola whether or not anyone taught you the word, your percentage means almost nothing at five attempts and something at a hundred, and the pause before you release is what sports science has spent thirty years on. Sport is not one subject. It is where about six of them turn out to be the same thing.
The point
Here is what the game was doing while you played it. Forty-five degrees is the best angle only when the target is level with your hands, and a hoop never is — which is why the answer sits near fifty-one, and why it moves the moment you turn on the leg drive. That is the opponent’s whole trick. The rest of the shot is the same story in other subjects: the force starts at the floor rather than the arm, the arc is a parabola whether or not anyone taught you the word, your percentage means almost nothing at five attempts and something at a hundred, and the pause before you release is what sports science has spent thirty years on. Sport is not one subject. It is where about six of them turn out to be the same thing.