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Projectile Motion

General

Projectile Motion (Physics)

The curved (parabolic) path traced by an object launched into the air and moving under gravity alone, with independent horizontal and vertical velocity components.

Definition

Projectile motion describes the path of any object launched into the air and moving under the influence of gravity alone — a thrown ball, a kicked football, or a fired artillery shell (ignoring air resistance). The defining insight is that horizontal and vertical motion are completely independent: horizontal velocity remains constant throughout the flight since no horizontal force acts on the object, while vertical velocity changes continuously due to gravity, producing the familiar curved (parabolic) trajectory.

The Projectile Motion Calculator computes maximum height, time of flight, and horizontal range from a launch speed and angle.

Formula

Horizontal velocity: vₓ = v × cos(θ) Vertical velocity: vᵧ = v × sin(θ) Time of flight: T = 2vᵧ ÷ g Maximum height: H = vᵧ² ÷ (2g) Range: R = vₓ × T

where v is launch speed, θ is launch angle, and g is gravitational acceleration (9.8 m/s² on Earth).

Worked Example

A ball is launched at 20 m/s at a 30-degree angle. Horizontal velocity: 20 × cos(30°) ≈ 17.3 m/s. Vertical velocity: 20 × sin(30°) = 10 m/s. Time of flight: 2 × 10 ÷ 9.8 ≈ 2.04 seconds. Maximum height: 10² ÷ (2 × 9.8) ≈ 5.1 meters. Range: 17.3 × 2.04 ≈ 35.3 meters.

Key Things to Know

  • Horizontal and vertical motion are independent: horizontal velocity stays constant; only vertical velocity is affected by gravity.
  • 45 degrees maximizes range for a given launch speed, when launch and landing heights are equal.
  • The vertical component matches free fall equations exactly, just combined with constant horizontal motion.
  • Real-world air resistance reduces both range and height compared to the idealized formulas used here, especially for light or fast-moving objects.
  • Velocity components (not just speed) drive the math, since direction determines how launch speed splits between horizontal and vertical motion.

Related Calculators

Projectile Motion Calculator →Calculate projectile motion instantly — enter initial velocity and launch angle to get max height, time of flight, and horizontal range, with a trajectory diagram.
Free Fall Calculator →Calculate free fall time and final velocity from a drop height under gravity. Enter a height in meters to instantly get fall time and impact velocity, ignoring air resistance.

Related Terms

Free FallFree Fall (Physics)VelocityVelocity (Physics)

Frequently Asked Questions

Horizontal and vertical motion are independent of each other during projectile motion — no horizontal force acts on the object (ignoring air resistance), so horizontal velocity stays constant throughout the flight, while vertical velocity changes continuously due to gravity. Splitting the motion into these two independent components makes the math tractable, since each obeys simple constant-velocity or constant-acceleration equations separately.
For a given launch speed, a 45-degree launch angle produces the maximum horizontal range (assuming launch and landing height are equal and ignoring air resistance), since it splits the initial velocity evenly between horizontal and vertical components. Angles above or below 45 degrees produce a shorter range, though pairs of complementary angles (like 30° and 60°) produce equal ranges to each other.
Yes — real-world projectiles like thrown balls or artillery shells experience air resistance (drag), which reduces both range and height compared to the idealized no-air-resistance formulas. The [Projectile Motion Calculator](/projectile-motion-calculator/) uses the standard idealized equations, which are a close approximation for dense, compact objects at moderate speeds but less accurate for light or high-speed objects.
Time of flight (for equal launch and landing height) is calculated as twice the time to reach maximum height: T = 2 × (v × sin(θ)) ÷ g, where v is launch speed, θ is launch angle, and g is gravitational acceleration. This comes directly from the vertical velocity component decelerating to zero at the peak, then symmetrically accelerating back down.
Free fall is a special case of vertical-only motion (no horizontal velocity component), while projectile motion generally involves both horizontal and vertical components together. The vertical component of projectile motion follows exactly the same equations as free fall — it's simply combined with independent horizontal motion to produce the full parabolic trajectory.