⬆️ Maximum Height Calculator – Projectile Motion
Calculate the maximum height reached by a projectile
How to Use This Calculator
Enter Initial Velocity
Input the initial speed at which the projectile is launched, measured in meters per second (m/s). This is the magnitude of the velocity vector.
Enter Launch Angle
Input the angle at which the projectile is launched above the horizontal, measured in degrees. For maximum range without air resistance, use 45°.
Enter Initial Height (Optional)
If the projectile starts from a height above ground level, enter that initial height in meters. Leave as 0 if launched from ground level.
Click Calculate
Press the "Calculate Maximum Height" button to compute the maximum height reached by the projectile, along with the time taken to reach that height.
Review Results
The calculator displays the maximum height, time to reach maximum height, and the vertical component of the initial velocity. Use these values for your physics calculations or analysis.
Formula
H_max = h₀ + (v₀² sin²(θ)) / (2g)
Where:
H_max = Maximum height (m)
h₀ = Initial height (m)
v₀ = Initial velocity (m/s)
θ = Launch angle (degrees)
g = Acceleration due to gravity (m/s²)
Example 1: Projectile launched from ground level
Given: Initial velocity = 30 m/s, Launch angle = 45°, Initial height = 0 m, Gravity = 9.81 m/s²
Step 1: Calculate vertical component
v_y = v₀ sin(θ) = 30 × sin(45°) = 30 × 0.707 = 21.21 m/s
Step 2: Calculate maximum height
H_max = 0 + (21.21²) / (2 × 9.81) = 450.32 / 19.62 = 22.94 m
Example 2: Projectile launched from a height
Given: Initial velocity = 25 m/s, Launch angle = 60°, Initial height = 10 m, Gravity = 9.81 m/s²
Step 1: Calculate vertical component
v_y = 25 × sin(60°) = 25 × 0.866 = 21.65 m/s
Step 2: Calculate height gained
Height gained = (21.65²) / (2 × 9.81) = 468.72 / 19.62 = 23.87 m
H_max = 10 + 23.87 = 33.87 m
Example 3: Time to reach maximum height
For the projectile in Example 1:
t_max = v_y / g = 21.21 / 9.81 = 2.16 seconds
At maximum height, the vertical velocity component becomes zero, and the projectile begins to fall back down.
About Maximum Height Calculator – Projectile Motion
The Maximum Height Calculator for Projectile Motion is an essential tool for physics students, engineers, and anyone working with projectile motion problems. This calculator determines the highest point reached by a projectile launched at an angle with an initial velocity, taking into account the initial height and gravitational acceleration. Understanding maximum height is crucial for analyzing trajectories in sports, ballistics, engineering applications, and physics education.
When to Use This Calculator
- Physics Homework: Solve projectile motion problems involving maximum height calculations
- Sports Analysis: Calculate the peak height of a basketball shot, football kick, or baseball hit
- Engineering Design: Determine maximum heights for projectiles in mechanical systems or safety calculations
- Ballistics: Analyze the trajectory peak height for bullets, arrows, or other projectiles
- Educational Purposes: Understand the relationship between launch angle, velocity, and maximum height
- Safety Assessments: Calculate maximum heights for objects launched accidentally or in emergency scenarios
Why Use Our Calculator?
- ✅ Accurate Calculations: Uses precise physics formulas based on kinematic equations
- ✅ Instant Results: Get maximum height and time calculations immediately without manual computation
- ✅ Handles Initial Height: Accounts for projectiles launched from elevated positions
- ✅ Educational Value: Shows the formula and step-by-step calculations for learning
- ✅ Multiple Outputs: Provides maximum height, time to reach it, and vertical velocity component
- ✅ Customizable Gravity: Adjust gravitational acceleration for different planets or scenarios
Common Applications
Sports Physics: Calculate the maximum height of a basketball shot to determine if it will clear the rim, or analyze the peak height of a soccer ball during a corner kick.
Engineering Projects: Determine the maximum height reached by projectiles in mechanical systems, such as launching mechanisms or automated throwing devices.
Safety Planning: Calculate maximum heights for fireworks, rockets, or other projectiles to ensure safe launch distances and clearances.
Physics Education: Help students understand how launch angle and initial velocity affect the maximum height reached by a projectile, demonstrating key concepts in kinematics.
Tips for Best Results
- Use Consistent Units: Ensure all inputs use meters (m) for distance and meters per second (m/s) for velocity
- Check Angle Range: Launch angles should be between 0° (horizontal) and 90° (vertical) for meaningful results
- Consider Air Resistance: This calculator assumes no air resistance; results will differ for real-world scenarios with significant air drag
- Initial Height Matters: Don't forget to include initial height if the projectile starts from an elevated position
- Verify Results: For a 45° launch angle, maximum height is approximately half the maximum range (without initial height)
Frequently Asked Questions
What is the maximum height for a projectile launched at 45°?
For a projectile launched from ground level at 45° with initial velocity v₀, the maximum height is H_max = v₀² / (4g). This is exactly half the maximum range (which occurs at 45°). For example, with v₀ = 30 m/s, H_max = 900 / (4 × 9.81) ≈ 22.94 m.
Does the maximum height depend on the launch angle?
Yes! Maximum height increases as the launch angle approaches 90° (vertical). For the same initial velocity, a projectile launched at 90° reaches the maximum possible height, while a projectile launched at 0° (horizontal) has zero maximum height gain. The relationship is H_max ∝ sin²(θ).
How does initial height affect maximum height?
Initial height is added directly to the height gained during flight. If you launch from 10 m above ground and gain 20 m during flight, your maximum height is 30 m. The formula is H_max = h₀ + (v₀² sin²(θ)) / (2g), where h₀ is the initial height.
At what point does the projectile reach maximum height?
The projectile reaches maximum height when its vertical velocity component becomes zero. This occurs at time t = v₀ sin(θ) / g. At this moment, the projectile is at the peak of its trajectory and begins to fall back down due to gravity.
What happens if air resistance is significant?
This calculator assumes no air resistance, which is valid for low speeds and dense objects. For high-speed projectiles or light objects, air resistance reduces both maximum height and range. The actual maximum height will be lower than calculated, and the trajectory will be asymmetric.
Can I use this for projectiles on other planets?
Yes! Simply change the gravity value. For example, use g = 1.62 m/s² for the Moon, g = 3.71 m/s² for Mars, or g = 24.79 m/s² for Jupiter. The calculator will automatically adjust the maximum height calculations accordingly.