Hit a Target
A basket, a window, a bin across the room: at what angle do you have to throw to hit it, and how fast? Or where does a given throw pass through? Solve for the angle, the speed or the target position — and drag the target around on the chart.
How Do You Aim a Throw?
For a fixed launch speed, most targets can be hit in two ways: a flat, fast arc and a high lob. The lob takes longer and comes down more steeply — that's why basketball players use a high arc: the ball drops into the hoop instead of skimming the rim.
The slower you throw, the closer the two arcs get, until at the minimum speed they merge into one. Every point you can reach at a given speed lies under a parabola called the reach boundary; a target outside it is out of reach, at any angle.
Trajectory
to scaleAiming Formulas
x, y from the launch point| Quantity | Formula |
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Worked Techniques
Rewrite the path equation so the only unknown is tan θ. The quadratic formula then gives both arcs at once; a negative discriminant means the target is out of reach.
At the minimum speed the two arcs merge. The best angle lies exactly halfway between straight up and the line of sight to the target (at angle φ).
If the angle is set — a cannon on a fixed mount, a ramp — solve the path equation for v₀ instead. You have to aim above the line of sight, otherwise no speed works.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 4 (Motion in Two and Three Dimensions).