T TibCal

Projectile Height Calculator

Calculate projectile height calculator calculate maximum apex height H from launch velocity v0 and angle theta (H = (v0 * sin(theta))^2 / (2 * g)). Accurate physics formulas and unit conversions for engineers, students & technicians.

Calculate Peak Apex Height (H)

Enter your physical parameters below to compute verified height metrics.

Initial launch speed in m/s (e.g. 60.0 m/s).
Launch angle relative to horizontal in degrees (e.g. 60.0°).

Calculation Results

Primary Metric Output --
Metric Breakdown 1 --
Metric Breakdown 2 --
Metric Breakdown 3 --
Metric Breakdown 4 --
Metric Breakdown 5 --
Mathematical Standard --

Calculated using verified physical methodology: Maximum Peak Height: H = \(v_0 \sin\theta)^2 / 2g
Apex Time: t_{apex} = \v_0 \sin\theta / g

*Note: Results represent peak apex elevation height above launch datum.

Quick Summary

The Projectile Height Calculator computes maximum apex elevation height ($H = \(v_0 \sin\theta)^2 / 2g$) reached by a projectile across meters, kilometers, and feet.

Formula Explanation

Maximum Peak Height: H = \(v_0 \sin\theta)^2 / 2g
Apex Time: t_{apex} = \v_0 \sin\theta / g

How It Works

The Projectile Height Calculator squares initial vertical velocity ($v_{0y} = v_0 \sin\theta$) and divides by twice Earth gravity ($2g = 19.6133\text{ m/s}^2$). It outputs maximum apex height in meters, km, and feet.

Step-by-Step Worked Example

Practical Problem: Calculate maximum height reached by a projectile launched at $v_0 = 60.0\text{ m/s}$ at an angle $\theta = 60.0^\circ$.

  1. Step 1: Identify Input Values: $v_0 = 60.0\text{ m/s}$, $\theta = 60.0^\circ$, $g = 9.80665\text{ m/s}^2$.
  2. Step 2: Calculate Vertical Velocity Component ($v_{0y}$): $v_{0y} = 60.0 \times \sin(60^\circ) = 60.0 \times 0.8660 = 51.96\text{ m/s}$.
  3. Step 3: Square Vertical Velocity: $v_{0y}^2 = (51.96)^2 = 2,700.0\text{ m}^2/\text{s}^2$.
  4. Step 4: Execute Height Division: $H = \2,700.0 / 2 \times 9.80665 = \2,700.0 / 19.6133 = 137.66\text{ meters}$.
  5. Step 5: Convert and Interpret Imperial & Metric Units: Peak Height $H = 137.66\text{ meters} = 0.1377\text{ km}$. Imperial Feet: $137.66 \times 3.28084 = 451.64\text{ feet}$. Time to Apex $t_{apex} = \51.96 / 9.80665 = 5.30\text{ seconds}$.

Real-World Calculation Examples

Scenario 1: High-Arc 60° Launch

Parameters: $v_0 = 60\text{ m/s}$, $\theta = 60^\circ$

Result: $H = 137.66\text{ m}$ (451.64 ft), $t_{apex} = 5.30\text{ s}$. 60° peak height.

Scenario 2: Direct Vertical Rocket Launch (90°)

Parameters: $v_0 = 100\text{ m/s}$, $\theta = 90^\circ$

Result: $H = 509.86\text{ m}$ (1,672.77 ft), $t_{apex} = 10.20\text{ s}$. Maximum vertical height.

Scenario 3: Football Punt Apex Height

Parameters: $v_0 = 25\text{ m/s}$, $\theta = 60^\circ$

Result: $H = 23.90\text{ m}$ (78.41 ft), $t_{apex} = 2.21\text{ s}$. Football punt apex.

Scenario 4: Flat Drive 15° Launch

Parameters: $v_0 = 50\text{ m/s}$, $\theta = 15^\circ$

Result: $H = 8.54\text{ m}$ (28.01 ft), $t_{apex} = 1.32\text{ s}$. Flat low-trajectory apex.

Key Benefits of Using This Calculator

Apex Height & Timing

Solves both maximum height ($H$) and exact time to reach peak apex ($t_{apex}$).

Vertical Velocity Zeroing

Verifies vertical velocity component drops to $v_y = 0\text{ m/s}$ at peak apex.

Multi-Unit Readouts

Outputs maximum height in meters, kilometers, and feet.

100% Free & Client-Side

Executes locally in your browser with zero latency or web server transmission.

Frequently Asked Questions (FAQ)

What is projectile maximum height?

Maximum height (H) is the highest vertical point (apex) reached by a projectile during its trajectory.

What is the formula for maximum height H?

H = (v0 * sin(theta))^2 / (2 * g) on level terrain.

What launch angle gives maximum height?

A launch angle of theta = 90 degrees (straight vertical launch) gives maximum height H_max = v0^2 / (2 * g).

What is vertical velocity at the highest point?

At peak apex, vertical velocity vy is zero (vy = 0 m/s); only horizontal velocity v0x remains.

How converts meters to feet?

Multiply meters by 3.28084 to obtain feet (e.g. 137.66 m = 451.64 ft).

How relates maximum height to potential energy?

At peak height, initial vertical kinetic energy converts fully to potential energy: 0.5 * m * v0y² = m * g * H.

How long does it take to reach maximum height?

t_apex = (v0 * sin(theta)) / g, which is exactly half of total flight time T on flat ground.

How does air resistance affect maximum height?

Air drag opposing upward motion reduces actual apex height compared to ideal vacuum calculations.

How does planetary gravity affect maximum height?

Height H is inversely proportional to gravity g; on the Moon (g = 1.62 m/s²), height is ~6x higher than on Earth for the same launch.

What is the SI unit of height?

The meter (m) is the fundamental SI unit of length and height.

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