Library / Mechanical / MEC-40
Mechanical
Projectile time of flight
Total time in the air for a drag-free projectile that lands at its launch height.
- Ref
- MEC-40
- Rev
- v1
- Field
- Mechanical
- Result
- T [s]
v is the launch speed, θ the launch angle and g the gravitational field strength. The projectile rises and falls over equal times if it lands at the height it left from, so T is twice the time to the apex, 2 v sin θ / g. Multiplying by the horizontal speed v cos θ gives the range.
Total time in the air for a drag-free projectile that lands at its launch height.
Calculator
- Horizontal range v² sin 2θ / g1,019 m
- Time to the highest point7.208 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The default shot is in the air for 14.4 s and travels about 1019 m in that time, an average horizontal speed of 70.7 m/s.
- The time of flight sets how long a target has to react or a guidance system has to act.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| v Launch speed | 1 to 2,000 m/s | 0.1442 | 288.3 | 288.2 | +10% | +10 % |
| g Gravitational field strength | 1 to 30 m/s² | 4.714 | 141.4 | 136.7 | +10% | −9.09 % |
| θ Launch angle | 1 to 89 ° | 0.3558 | 20.38 | 20.03 | +10% | +7.54 % |
Sweep of Launch speed
| Launch speed (m/s) | Time of flight (s) |
|---|---|
| 1 | 0.1442 |
| 2.586 | 0.3728 |
| 6.687 | 0.9641 |
| 17.29 | 2.493 |
| 44.72 | 6.447 |
| 115.6 | 16.67 |
| 299.1 | 43.11 |
| 773.4 | 111.5 |
| 2,000 | 288.3 |
Sweep one variable
Variables and how each one contributes
Time of flight
What the equation gives you. Everything else feeds it.
Launch speed
The launch speed. Directly proportional: double the speed, double the time in the air.
Launch angle
The launch angle above the horizontal. Acts through sin θ: the time rises with angle up to 90°. A 10 % rise in θ from 45° (to 49.5°) raises T by about 7.5 %.
Gravitational field strength
The gravitational field strength (9.81 m/s² on Earth). Inverse: halving g doubles the time of flight.
Worked example with the values above
- Vertical part of the speed: v sin θ = 100 × sin 45° = 70.71 m/s.
- Double it: 141.4.
- T = 141.4 / 9.81 = 14.42 s.
Assumptions and limits
- No air resistance.
- The projectile lands at the height it was launched from, on level ground.
Common mistakes
- Using the time to the apex as the time of flight. It is half the answer.
- Applying it when the target is at a different height, which needs the full quadratic solution.
Statistics and uncertainty
T is proportional to v sin θ / g, so errors in the speed and in g pass straight through while the angle error is weighted by cot θ (equal to 1 at 45°). Drag shortens the real time of flight and makes the descent steeper than the climb.
Where it is used
- Time available to a target to react, or to a guidance system to act.
- Combining with horizontal speed to find range.
Maths you need
- Trigonometry: v sin θ
- Constant acceleration: v = u + at