Library / Mechanical / MEC-39
Mechanical
Projectile maximum height
Peak height of a drag-free projectile.
- Ref
- MEC-39
- Rev
- v1
- Field
- Mechanical
- Result
- H [m]
v is the launch speed, θ the launch angle above the horizontal and g the gravitational field strength. Only the vertical component v sin θ matters for the peak height: it is brought to zero by gravity over a rise of (v sin θ)²/2g. The default is a 100 m/s launch at 45°.
Peak height of a drag-free projectile. Only the vertical part of the launch speed matters.
Calculator
- Vertical launch speed v sin θ70.71 m/s
- 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.
- At 45° the shot reaches 255 m, exactly half of the 510 m it would reach fired straight up at the same speed.
- The rise and the fall take equal time, so the highest point is reached at half the time of flight.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| v Launch speed | 1 to 2,000 m/s | 0.02548 | 101,900 | 101,900 | +10% | +21 % |
| g Gravitational field strength | 1 to 30 m/s² | 83.33 | 2,500 | 2,417 | +10% | −9.09 % |
| θ Launch angle | 1 to 89 ° | 0.1552 | 509.5 | 509.4 | +10% | +15.6 % |
Sweep of Launch angle
| Launch angle (°) | Maximum height (m) |
|---|---|
| 1 | 0.1552 |
| 12 | 22.03 |
| 23 | 77.81 |
| 34 | 159.4 |
| 45 | 254.8 |
| 56 | 350.3 |
| 67 | 431.9 |
| 78 | 487.7 |
| 89 | 509.5 |
Sweep one variable
Variables and how each one contributes
Maximum height
What the equation gives you. Everything else feeds it.
Launch speed
The launch speed. Squared: doubling the speed quadruples the height. A 10 % rise in v raises H by 21 %.
Launch angle
The launch angle above the horizontal. Acts through sin²θ. Height rises steeply from 0° and flattens near 90°, where it reaches v²/2g. At 45° a 10 % rise in θ (to 49.5°) raises H by about 16 %.
Gravitational field strength
The gravitational field strength (9.81 m/s² on Earth). Inverse: on the Moon (1.62 m/s²) the same shot goes six times higher.
Worked example with the values above
- Vertical part of the speed: v sin θ = 100 × sin 45° = 70.71 m/s.
- Square it: 5,000; double g: 19.62.
- H = 5,000 / 19.62 = 254.8 m.
Assumptions and limits
- No air resistance, so only gravity acts.
- A flat Earth with constant g, and a launch and apex at similar altitudes.
Common mistakes
- Forgetting to square the sine as well as the speed.
- Using a calculator in radian mode when θ is in degrees.
Statistics and uncertainty
H ∝ v² sin²θ, so a 1 % error in the speed gives a 2 % error in the height. Near 90° the height barely changes with angle, so angle errors matter little there. Real drag reduces the true height by a large factor for a fast shell, and the error grows with speed.
Where it is used
- Apex height of a ballistic trajectory, to check for obstacles.
- Finding the ceiling needed for a mortar or thrown-object path.
Maths you need
- Trigonometry: v sin θ as the vertical component
- Constant acceleration: v² = u² + 2as