The path of a thrown object
Once launched, a ball, a stone or a jet of water follows a parabola. Its horizontal speed stays constant, while gravity steadily pulls its vertical speed down at 9.81 m/s² (32.2 ft/s²). Splitting the launch velocity into those two parts gives every property of the flight. The calculator neglects air resistance, which is reasonable for heavy, compact objects at moderate speeds.
The formulas
With launch speed v₀, angle θ and launch height h₀:
- Horizontal speed vₓ = v₀ × cos θ; vertical speed v_y = v₀ × sin θ
- Time of flight t = (v_y + √(v_y² + 2g h₀)) ÷ g
- Range = vₓ × t
- Maximum height = h₀ + v_y² ÷ (2g), reached after v_y ÷ g seconds
- From ground level, the range simplifies to v₀² × sin(2θ) ÷ g, largest at 45°.
Worked example: 45 mph at 45° from the ground
- 45 mph = 20.1 m/s, so vₓ = v_y = 20.1 × 0.707 = 14.2 m/s.
- Time of flight: 2 × 14.2 ÷ 9.81 = 2.90 s.
- Range: 14.2 × 2.90 = 41.3 m, or 135 ft.
- Maximum height: 14.2² ÷ 19.61 = 10.3 m, or 34 ft, reached after 1.45 s.
- The ball lands at 45 mph and 45° below the horizontal, the mirror image of the launch.
Range at other angles (45 mph, from the ground)
| Angle | Range | Max height | Flight time |
|---|---|---|---|
| 15° | 68 ft | 4.6 ft | 1.06 s |
| 30° | 117 ft | 17 ft | 2.05 s |
| 45° | 135 ft | 34 ft | 2.90 s |
| 60° | 117 ft | 51 ft | 3.55 s |
| 75° | 68 ft | 63 ft | 3.96 s |
Complementary angles such as 30° and 60° give the same range from the ground: the higher shot just stays longer in the air.
Launching from a height
From a raised point, the best angle is a little below 45°, because the projectile has extra time to travel forward while it falls. A shot put released at 7 ft is best thrown at about 37 to 42°, depending on speed.
Where air resistance changes the answer
Drag shortens the range and makes the descent steeper than the climb. A baseball hit at 100 mph would fly over 650 ft in a vacuum but only about 400 ft in real air; spin adds lift and changes it again. Use the calculator for physics problems, garden hoses, slow throws and as an upper bound for fast objects.
Practical uses
- Sports: estimating how speed and angle trade off for a throw, a kick or a jump.
- Water jets: a fountain or a hose follows the same parabola; the exit speed comes from the pressure.
- Safety distances: the maximum range at 45° gives the upper bound of where debris or a thrown object can land.