Trajectory Calculator
Calculate and visualize the parabolic path of projectile motion
Calculate Projectile Trajectory
Speed at which projectile is launched
Angle above horizontal (45° gives maximum range at ground level)
Height above ground level (0 = ground level)
m/s² (Earth: 9.81, Moon: 1.62, Mars: 3.71)
Find projectile height at specific horizontal distance
Trajectory Formula
Trajectory Analysis Results
Trajectory type: Ground level launch
Input values: V₀ = 20.00 m/s, α = 45.0°, h₀ = 0.00 m
Trajectory shape: Parabolic path (quadratic equation) - air resistance neglected
Trajectory Visualization
Physics Analysis
Example Calculations
Water Fountain Example
Initial velocity: 5 ft/s (1.52 m/s)
Launch angle: 60°
Initial height: 5 in (0.127 m)
Expected result: Small parabolic arc typical of fountain jets
Projectile at 30° and 10 m/s
y = x × tan(30°) - (9.81 × x²) / (2 × 10² × cos²(30°))
y = x × 0.577 - (9.81 × x²) / (2 × 100 × 0.75)
y = 0.577x - 0.0654x²
Maximum range: ≈ 8.8 meters
Basketball Shot
Initial velocity: 7 m/s
Launch angle: 50° (typical basketball shot)
Initial height: 2 m (player's release point)
Target: Basketball hoop at 3.05m height, ~4.5m distance
Trajectory Components
Initial Velocity
Launch speed and direction
Determines energy and trajectory shape
Launch Angle
Angle above horizontal
45° optimal for max range (level ground)
Initial Height
Launch elevation
Extends range and flight time
Gravity
Downward acceleration
Creates parabolic path
Common Trajectories
Physics Tips
Trajectory is always parabolic under constant gravity
45° gives maximum range for ground-level launch
Higher launch angles increase flight time
Horizontal velocity remains constant (no air resistance)
Peak occurs at half the total flight time
Understanding Projectile Trajectory
What is Trajectory?
Trajectory is the path followed by a moving object under the action of gravity. For projectiles, this path is parabolic due to the constant downward acceleration of gravity combined with constant horizontal velocity.
Key Characteristics
- •Parabolic shape (quadratic equation)
- •Symmetric about peak point
- •Independent horizontal and vertical motions
- •Constant horizontal velocity
- •Accelerated vertical motion
Trajectory Formula Derivation
Step 1: Motion Equations
x = V₀ₓ × t = V₀ × cos(α) × t
y = h₀ + V₀ᵧ × t - ½gt²
Step 2: Eliminate Time
t = x / (V₀ × cos(α))
Step 3: Final Formula
y = h₀ + x×tan(α) - gx²/(2V₀²cos²(α))
Real-World Applications
Sports
- • Basketball shooting
- • Golf ball trajectories
- • Baseball/softball throws
- • Soccer ball kicks
- • Javelin throwing
Engineering
- • Water fountain design
- • Ballistics calculations
- • Missile trajectories
- • Projectile launchers
- • Safety systems
Science
- • Ballistics research
- • Physics education
- • Astronomy (satellite orbits)
- • Forensic analysis
- • Video game physics
Optimal Launch Angles
Maximum Range
For ground-level targets, 45° provides maximum horizontal distance.
R_max = V₀² / g (at α = 45°)
Maximum Height
For maximum altitude, use 90° (vertical launch).
H_max = V₀² / (2g) (at α = 90°)