Hong Life CalculatorsUnderstanding the projectile motion — concept and principle
A projectile is any object that, after launch, moves only under gravity: a thrown ball, a jet of water, a stone kicked off a ledge. This calculator assumes no air resistance and returns the horizontal range, flight time, maximum height and impact speed from the launch speed, launch angle, launch height and gravitational acceleration. It is used in physics lessons, for sports intuition and for quick first estimates.
How it is calculated
- Split the launch velocity: vₓ = v cos θ (horizontal) and v_y = v sin θ (vertical).
- Flight time: t = (v_y + √(v_y² + 2gh₀)) ÷ g, the positive root of h₀ + v_y t − ½gt² = 0, i.e. when the projectile reaches ground level.
- Range: R = vₓ × t.
- Maximum height: H = h₀ + v_y² ÷ (2g).
- Impact speed: √(vₓ² + (v_y − gt)²).
Units: m/s, degrees, m and m/s². The default g = 9.80665 m/s² is the standard gravity value.
The principle
Without air resistance, gravity acts only vertically, so the two directions are independent. Horizontally there is no force and vₓ stays constant; vertically the motion is uniformly accelerated with acceleration −g. Combining x = vₓt with y = h₀ + v_y t − ½gt² gives a parabola, as Galileo showed. The top of the path is where the vertical velocity becomes zero, at t = v_y ÷ g. Energy conservation also explains the impact speed: falling back to launch height returns the original speed, and falling further adds speed so that v_impact² = v² + 2gh₀.
Worked example
- Inputs: v = 20 m/s, θ = 45°, h₀ = 0 m, g = 9.80665 m/s².
- vₓ = v_y = 20 × cos 45° ≈ 14.1421 m/s.
- With h₀ = 0, t = 2v_y ÷ g = 28.2843 ÷ 9.80665 ≈ 2.8842 s.
- Range = 14.1421 × 2.8842 ≈ 40.7886 m.
- Maximum height = 14.1421² ÷ (2 × 9.80665) = 200 ÷ 19.6133 ≈ 10.1972 m.
- Vertical velocity at landing = 14.1421 − 9.80665 × 2.8842 = −14.1421 m/s, so impact speed = 20 m/s.
| Launch speed (m/s) | Range (m) |
|---|---|
| 5 | 2.549290532 |
| 10 | 10.19716213 |
| 15 | 22.94361479 |
| 20 | 40.78864852 |
| 25 | 63.73226331 |
| 30 | 91.77445917 |
| 40 | 163.1545941 |
How to read the result
On level ground the range is R = v² sin 2θ ÷ g, greatest at 45° (here 40.79 m), and complementary angles such as 30° and 60° give the same range with different heights and times. Launching from a height (h₀ > 0) makes the best angle slightly less than 45°. Doubling the speed quadruples the range and maximum height.
Common mistakes
- Entering the angle in radians; the field expects degrees.
- Expecting real balls to match: drag, spin and wind usually shorten the range considerably.
- Forgetting that with h₀ > 0 the range and flight time are measured to ground level (height 0), not back to launch height.
Frequently asked questions
What angle gives maximum height?
90°, straight up, giving H = v² ÷ (2g) but zero range.
How do I use this on the Moon?
Enter g ≈ 1.62 m/s². With the same speed and angle the range is about six times longer.
Does the mass of the projectile matter?
Not without air resistance; all objects fall with the same acceleration.
Sources: Halliday, D., Resnick, R. & Walker, J. Fundamentals of Physics (Wiley). · BIPM, The International System of Units (SI Brochure) — standard acceleration of gravity 9.80665 m/s².