Maximum Height Calculator – Projectile Motion
Calculate the maximum height reached by projectiles in motion
Calculate Maximum Height
Speed at which projectile is launched
Angle above horizontal
Height from which projectile is launched
Projectile Motion Results
Physics Formula
Maximum Height: h_max = h₀ + v_y² / (2g)
Where:
- • h₀ = initial height (0.00 m)
- • v_y = vertical velocity component (21.21 m/s)
- • g = gravitational acceleration (9.807 m/s²)
Calculation: 0.00 + (21.21)² / (2 × 9.807) = 22.94 m
Physics Analysis
Example: Soccer Ball Kick
Problem Setup
Scenario: A soccer ball is kicked at 30 ft/s at a 70° angle
Given: v₀ = 30 ft/s, α = 70°, h₀ = 0 ft
Question: Can the ball clear a 13 ft fence?
Solution Steps
1. Convert to SI units: v₀ = 30 × 0.3048 = 9.144 m/s
2. Find vertical component: v_y = 9.144 × sin(70°) = 8.59 m/s
3. Calculate max height: h_max = 0 + (8.59)² / (2 × 9.807) = 3.76 m
4. Convert back to feet: 3.76 m = 12.35 ft
Result: Ball reaches 12.35 ft - won't clear 13 ft fence!
Real-World Examples
Basketball Shot
Basketball free throw
Soccer Ball Kick
Soccer ball kick from ground
Baseball Throw
Baseball throw from pitcher
Cannonball Launch
Historical cannon shot
Key Physics Concepts
Maximum Height
Highest point in projectile trajectory
Velocity Components
Horizontal and vertical velocity parts
Gravity
Downward acceleration (9.807 m/s²)
Launch Angle
Angle above horizontal direction
Essential Formulas
Maximum Height
h_max = h₀ + v_y² / (2g)
Height including initial elevation
Vertical Velocity
v_y = v₀ sin(α)
Upward velocity component
Time to Max Height
t = v_y / g
Time when v_y becomes zero
Total Flight Time
t = 2v_y / g (level ground)
Time to return to ground
Understanding Maximum Height in Projectile Motion
What is Maximum Height?
Maximum height is the highest vertical position reached by a projectile during its flight. At this point, the vertical velocity becomes zero before the projectile starts falling back down.
Physics Behind the Calculation
The maximum height formula comes from energy conservation and kinematics. The vertical velocity component provides the initial kinetic energy that gets converted to potential energy at the peak.
Key Factors
Initial Velocity
Higher velocity = greater maximum height
Launch Angle
90° gives maximum height for given velocity
Initial Height
Starting elevation adds to maximum height
Gravity
Constant downward acceleration
Real-World Applications
Sports
Basketball shots, soccer kicks, javelin throws
Engineering
Water fountains, sprinkler systems
Military
Artillery, missile trajectories
Safety
Obstacle clearance calculations
Important Principles
Energy Conservation
Kinetic energy converts to potential energy
Symmetry
Time up equals time down (level ground)
Independence
Horizontal and vertical motions are independent
Zero Velocity
Vertical velocity = 0 at maximum height