Maximum Height Projectile Motion Calculator
Free calculate peak height using h = v²sin²(θ)/(2g). Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Earth: 9.81 m/s²
Results
Enter parameters to calculate
What is Maximum Height in Projectile Motion?
The maximum height (H) is the highest point reached by a projectile during its flight. At this point, the vertical component of velocity becomes zero (v_y = 0).
Maximum height depends on the initial velocity and launch angle. For a given initial speed, maximum height occurs at a 90° launch angle (straight up). At 45°, the projectile achieves maximum range, but not maximum height.
The maximum height is reached when the vertical velocity component equals zero. This occurs at time t = v₀sin(θ)/g, and the height is H = v₀²sin²(θ)/(2g).
Maximum Height Formula
Where:
- • H = Maximum height (m)
- • v₀ = Initial velocity (m/s)
- • θ = Launch angle (degrees)
- • g = Acceleration due to gravity (m/s²)
Special cases:
θ = 90°: H = v₀²/(2g) (maximum possible height)
θ = 45°: H = v₀²/(4g) (half of maximum height)
How to Calculate
-
1
Convert angle to radians (if needed)
For calculations: θ_rad = θ_deg × π/180. Most calculators work with degrees directly.
-
2
Calculate sin(θ)
Find the sine of the launch angle.
-
3
Square the sine
Calculate sin²(θ) = (sin(θ))².
-
4
Calculate maximum height
H = v₀²sin²(θ)/(2g). Square velocity, multiply by sin²(θ), divide by 2g.
Practical Examples
Example 1: 45° Launch
v₀ = 20 m/s, θ = 45°, g = 9.81 m/s².
Solution:
H = 20² × sin²(45°) / (2 × 9.81)
H = 400 × 0.5 / 19.62
H ≈ 10.19 m
Example 2: Vertical Launch
v₀ = 30 m/s, θ = 90°, g = 9.81 m/s².
Solution:
H = 30² × sin²(90°) / (2 × 9.81)
H = 900 × 1 / 19.62
H ≈ 45.87 m
Applications
Ballistics
Calculating maximum altitude for projectiles, missiles, and artillery. Understanding trajectory characteristics.
Sports
Analyzing maximum height of thrown balls, kicked footballs, and other projectiles in sports.
Fireworks
Designing firework displays, calculating peak heights, and ensuring safety clearances.
Education
Teaching kinematics, understanding projectile motion, and demonstrating physics principles.
Frequently Asked Questions
What angle gives maximum height?
90° (straight up) gives maximum height for a given initial speed. At 90°, sin(90°) = 1, so H = v₀²/(2g) is maximum. However, this gives zero range.
Why is maximum height at the midpoint?
At maximum height, vertical velocity is zero. The projectile has used all its upward velocity to overcome gravity. This occurs halfway through the flight time (for symmetric trajectories).
Does air resistance affect maximum height?
Yes! Air resistance reduces maximum height. The formula H = v₀²sin²(θ)/(2g) assumes no air resistance. In reality, drag reduces both height and range.
How does initial velocity affect height?
Height is proportional to v₀² - doubling velocity quadruples maximum height! This is because kinetic energy (½mv²) is converted to potential energy (mgh).
What's the time to reach maximum height?
t = v₀sin(θ)/g. This is when vertical velocity becomes zero. Total flight time is 2t (for symmetric trajectory landing at same height).