Projectile Motion Calculator - Velocity, Trajectory, Range & Apex

Calculate kinematic projectile trajectories including horizontal range, maximum flight apex height, total flight duration, impact velocity, and launch angles.

⚙️ Configuration & Inputs

Initial Launch Velocity (v₀) Velocity magnitude at launch

Velocity Unit

Launch Angle θ (degrees) Angle above horizontal plane (45° = max range)

Initial Elevation h₀ (m) Height above landing plane

Gravitational Field (g)

📊 Results & Analysis

Horizontal Range (Distance) 206.5 m 677.5 ft (0.128 miles)

Maximum Apex Height (H_max) 51.63 m 169.4 ft above ground

Total Flight Time (Duration) 6.49 sec Time to Peak: 3.24 sec

Impact Velocity 45.00 m/s Impact Angle: -45.0°

Horizontal Velocity (v_0x) 31.82 m/s Constant across trajectory (no drag)

Initial Vertical Velocity (v_0y) 31.82 m/s 71.18 mph upward

Kinematic Projectile Motion Equations

Projectile motion occurs when an object is given an initial velocity and follows a curved parabolic trajectory dictated solely by gravity (neglecting atmospheric drag).

1. Velocity Vector Decomposition

v_0x = v₀ × cos(θ) (horizontal component remains constant)
v_0y = v₀ × sin(θ) (vertical component decelerates at g)

2. Key Trajectory Formulas (Flat Ground h₀ = 0)

Time of Flight: t_total = (2 × v₀ × sin(θ)) / g
Peak Apex Height: H_max = (v₀² × sin²(θ)) / (2g)
Maximum Range: R = (v₀² × sin(2θ)) / g

Notice that maximum horizontal distance is achieved when sin(2θ) = 1, which occurs at an angle of θ = 45°.

Frequently Asked Questions

The horizontal range formula includes sin(2θ). The sine function reaches its maximum possible value of 1.0 when its argument is 90 degrees. Solving 2θ = 90° gives θ = 45° on flat terrain.

When launching from a cliff or elevated platform above the landing plane, the optimal angle is slightly below 45° (typically 38° to 43° depending on elevation) because the extra drop time allows horizontal velocity to travel further.

In ideal vacuum kinematics, mass cancels out completely from the equations (all masses accelerate identically under gravity at rate g). In the real atmosphere, heavier dense objects travel further because air drag decelerates them less relative to their momentum.