Maximum height H (m)
5.09858
Time of flight T (s)
2.03943
Horizontal velocity uₓ (m/s)
17.3205
Initial vertical velocity u_y (m/s)
10
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📘 About the Projectile Motion
A projectile is anything thrown or launched that then moves under gravity alone. This projectile motion calculator takes the launch speed and angle and gives the horizontal range, the maximum height and the time of flight.
It assumes the projectile lands at the same height it was launched from and ignores air resistance, as in Class 11 physics problems. The velocity is split into a horizontal part u cos θ, which stays constant, and a vertical part u sin θ, which gravity slows down.
🧮 Formula
The range is greatest at 45°. Angles that add up to 90° (such as 30° and 60°) give the same range.
✏️ Worked examples
Ball kicked at 20 m/s and 45°
You get| Range R (m) | 40.7886 |
| Maximum height H (m) | 10.1972 |
| Time of flight T (s) | 2.88419 |
| Horizontal velocity uₓ (m/s) | 14.1421 |
| Initial vertical velocity u_y (m/s) | 14.1421 |
The ball travels 40.7886 m, rises 10.1972 m and stays in the air for 2.88419 s.
Try this example →Same speed at 30°
You get| Range R (m) | 35.324 |
| Maximum height H (m) | 5.09858 |
| Time of flight T (s) | 2.03943 |
| Horizontal velocity uₓ (m/s) | 17.3205 |
| Initial vertical velocity u_y (m/s) | 10 |
At 30° the range is 35.324 m, the same as at 60°, but the flight is lower and shorter.
Try this example →🧭 How to use this calculator
What you enter
- Launch speed (u) (m/s)
- Launch angle (θ) (degrees)Angle above the horizontal, from 0 to 90.
- Acceleration due to gravity (g) (m/s²)
What you get
- Range R (m)
- Maximum height H (m)
- Time of flight T (s)
- Horizontal velocity uₓ (m/s)
- Initial vertical velocity u_y (m/s)
💡 Why 45° gives the maximum range
Range depends on sin 2θ, which is largest (equal to 1) when 2θ = 90°, so θ = 45°. Then R = u² ÷ g. In real life, air resistance makes the best angle a little lower, around 35°–43° for balls.
❓ Frequently asked questions
What is the formula for the range of a projectile?
R = u² sin 2θ ÷ g, for a projectile that lands at the height it was launched from.
At what angle is the range maximum?
At 45°, where R = u² ÷ g.
What is the horizontal velocity of a projectile?
u cos θ. It stays the same throughout the flight because, without air resistance, no horizontal force acts.
What is the velocity at the highest point?
Only the horizontal part, u cos θ. The vertical velocity is zero at the top.