Calculate the range w of an oblique projectile from launch speed v₀, launch angle α and launch height h above the landing point – or conversely the launch speed required for a given range. The trajectory is a parabola and the range follows from its root.
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Calculate the range w of an oblique projectile from launch speed v₀, launch angle α and launch height h above the landing point – or conversely the launch speed required for a given range. The trajectory is a parabola and the range follows from its root.
A shot is launched from h = 2.00 m at α = 44° with v₀ = 14.05 m/s. This gives w = 14.05²/(2·9.81)·sin 88° + √[14.05⁴/(4·9.81²)·sin²88° + 2·2·14.05²/9.81·cos²44°] ≈ 22.0 m. The same example from Dankert also shows the angle dependence: at α = 45° the range drops slightly to 21.95 m, and the maximum of 22.03 m occurs at α = 42.41° – not at 45°, because the launch point lies above the landing point.
Idealised point mass in a uniform gravitational field without air resistance, lift, Magnus effect from spin, or wind. Launch is instantaneous from a single point; body extent, the launching motion and rotation are not modelled. For compact dense bodies at moderate speeds this idealisation is usable; for balls, shuttlecocks and high speeds it is not – there air resistance shortens the range considerably and lowers the optimum launch angle further. The mathematical model has two roots; only the physically relevant positive range in the launch direction is reported.
In oblique projectile motion the launch angle is free – and that changes everything: the trajectory becomes a parabola, and the frequently quoted optimum of 45° holds only in one special case. This calculator determines the range from launch speed, angle and launch height, and can be rearranged for the required launch speed.
A body launched obliquely performs two independent motions at once: a uniform horizontal motion at v₀·cos α and a uniformly decelerated vertical motion with initial speed v₀·sin α. Eliminating time from both displacement-time laws gives the trajectory y = −g·x²/(2·v₀²·cos²α) + x·tan α + h – a downward-opening parabola. The range w is its root, i.e. the horizontal distance at which the trajectory reaches the impact plane y = 0.
Like a garden hose: held flat, the jet lands close. Held steeply upwards, the water goes high but likewise not far. The optimum lies somewhere in between – and holding the hose above your head instead of at ground level shifts that optimum to a flatter angle.
w = v₀²/(2g)·sin 2α + √[v₀⁴/(4g²)·sin²2α + 2h·v₀²/g·cos²α]
Trajectory: y = −g·x²/(2·v₀²·cos²α) + x·tan α + hRange: w = v₀²/(2g)·sin 2α + √[v₀⁴/(4g²)·sin²2α + 2h·v₀²/g·cos²α]Launch speed: v₀ = w·√(g/(w·sin 2α + 2h·cos²α))Special case of equal heights (h = 0): w = v₀²·sin 2α/g, maximal at α = 45°Special case of horizontal launch (α = 0): w = v₀·√(2h/g)| Symbol / input | Meaning |
|---|---|
| Range w | The horizontal distance travelled from the launch point until impact at the level h below it. For h = 0 (launch and landing at the same height) the result simplifies to the familiar w = v₀²·sin 2α/g. |
| Launch speed v₀ | The magnitude of the velocity on leaving the hand or the launching device, measured along the launch direction – not merely its horizontal component. The horizontal component is v₀·cos α, the vertical one v₀·sin α. Since v₀ enters squared, a 10 % higher launch speed does not double the range but increases it by about 21 %. |
| Launch angle α | The angle of the launch direction against the horizontal, positive upwards. α = 0° is horizontal projectile motion, α = 90° a vertical throw upwards (the range is then zero). With launch and landing at the same height the greatest range occurs exactly at 45°; if the launch point is higher than the landing point the optimum angle is below 45° – about 42.4° in the example. |
| Launch height h | The vertical height of the launch point above the plane on which impact is counted. h = 0 means launch and landing at the same height. Additional launch height extends the flight time and therefore the range, and it shifts the optimum launch angle downwards. |
| Gravitational acceleration g | The local gravitational acceleration; it is nearly the same everywhere on Earth. Typical value 9.81 m/s², standard value 9.80665 m/s². Adjust only for very precise calculations or for other celestial bodies. |
v₀ is the magnitude of the launch velocity along the launch direction, not merely its horizontal component. α is the launch angle against the horizontal, positive upwards. h is the height of the launch point above the plane on which impact is counted – at h = 0 launch and landing are at the same height. g is the gravitational acceleration, typically 9.81 m/s² or the standard value 9.80665 m/s².
Enter launch speed, angle and launch height; the result is the horizontal range. For the inverse – what launch speed a required range needs – select v₀ as the target and enter the desired range. The angle reference matters: it is measured against the horizontal, not the vertical. To find the best angle, vary α at fixed v₀ and h in one-degree steps and read off the maximum – analytically that optimum is only obtainable with considerable effort when h > 0.
A shot launched from h = 2.00 m at α = 44° with v₀ = 14.05 m/s reaches w ≈ 22.0 m. Worked backwards: a 22 m range at 44° from 2 m height requires v₀ = 14.05 m/s. The angle variation at unchanged 14.05 m/s is revealing: at 45° the range drops to 21.95 m, and the maximum of 22.03 m occurs at 42.41°. The optimum launch angle is therefore below 45° – precisely because the launch point lies 2 m above the landing point. Setting h = 0 makes the formula return exactly the classical relation w = v₀²·sin 2α/g with its maximum at exactly 45°.
Launch speed enters squared: 10 % more v₀ gives about 21 % more range, whereas 10 % more launch height extends the range only slightly. For throwing and slinging processes, speed is therefore almost always worth more than height. The angle dependence is very flat near the optimum – between 42° and 45° the example differs by only 0.08 m. That is reassuring in practice: small angular deviations cost hardly any range, while large ones do, because the curve falls off steeply towards 0° and 90°.
The calculation runs in metres, seconds and radians internally. Speed may be entered in m/s, km/h, ft/s, mph, m/min or mm/min, range and launch height in mm, cm, m, km or inches, and the angle in degrees or radians. Gravitational acceleration is given in m/s² or ft/s².
Discharge and throwing trajectories on conveyors, throwing blades and spreaders; launch ranges in sport and engineering (shot put, throwing events); positioning discharge points on belt conveyors and chutes; ballistic estimates at low speeds; safety distances for parts thrown clear; checking whether a catch bin or protected area is sized generously enough.
Idealised point mass in a uniform gravitational field without air resistance, lift, Magnus effect from spin, or wind. Launch is instantaneous from a single point; body extent, the launching motion and rotation are not modelled. This is usable for compact dense bodies at moderate speeds – but not for balls, shuttlecocks, paper or high speeds: there air resistance shortens the range considerably and lowers the optimum launch angle further. Mathematically the model has two roots; only the positive range in the launch direction is reported.
Common mistake: The most common error is assuming 45° is always optimal – that holds only for h = 0. Second, the horizontal velocity component is often entered instead of the launch speed itself, which underestimates the range. Third, the angle is measured against the vertical, which swaps sine for cosine. And fourth, the result is treated as reliable for light bodies such as balls although air resistance can shorten the range by tens of percent – which is conservative for safety distances but substantially too far for a target prediction.
Discharge and throwing trajectories on conveyors, throwing blades and spreaders; launch ranges in sport and engineering (shot put, throwing events); positioning discharge points on belt conveyors and chutes; ballistic estimates at low speeds; safety distances for parts thrown clear; checking whether a catch bin or protected area is sized generously enough.
v₀ is the magnitude of the launch velocity along the launch direction, not merely its horizontal component. α is the launch angle against the horizontal, positive upwards. h is the height of the launch point above the plane on which impact is counted – at h = 0 launch and landing are at the same height. g is the gravitational acceleration, typically 9.81 m/s² or the standard value 9.80665 m/s².
Idealised point mass in a uniform gravitational field without air resistance, lift, Magnus effect from spin, or wind. Launch is instantaneous from a single point; body extent, the launching motion and rotation are not modelled. This is usable for compact dense bodies at moderate speeds – but not for balls, shuttlecocks, paper or high speeds: there air resistance shortens the range considerably and lowers the optimum launch angle further. Mathematically the model has two roots; only the positive range in the launch direction is reported.