Home › Projectile Motion

Projectile Motion — Range, Height & Time of Flight

Projectile motion describes the curved path of any object launched into the air under gravity. This page covers how to separate horizontal and vertical components, all key formulas for range, height, and time of flight, and step-by-step worked examples.
Study Notes
Formulas
MCQ Practice

Study Notes: Projectile Motion

What is Projectile Motion?

Projectile motion occurs when an object is launched into the air and moves under the influence of gravity alone (ignoring air resistance). The defining characteristic is that the horizontal and vertical components of motion are completely independent of each other.

  • Horizontal motion: Constant velocity (no horizontal force acts). x = u cosθ × t
  • Vertical motion: Constant downward acceleration due to gravity. g = 9.81 m/s² downward.

Key Quantities

Time of flight (T): The total time the projectile is in the air. T = 2u sinθ/g. This is twice the time to reach maximum height.

Maximum height (H): The highest point reached. H = u² sin²θ / (2g). Maximum height occurs at θ = 90°.

Horizontal range (R): The horizontal distance covered. R = u² sin(2θ) / g. Maximum range occurs at θ = 45°.

Step-by-Step Method

  1. Resolve initial velocity into horizontal (u⊂x; = u cosθ) and vertical (u⊂y; = u sinθ) components.
  2. Apply SUVAT equations to the vertical component (a = −g = −9.81 m/s²).
  3. Apply constant velocity equations to the horizontal component (a = 0).
  4. Use t as the connecting variable between horizontal and vertical calculations.

Worked Example

A ball is kicked at 25 m/s at 30° above the horizontal. Find: (a) time of flight, (b) maximum height, (c) range. (g = 10 m/s²)

Resolve: u⊂x; = 25 cos30° = 21.7 m/s   u⊂y; = 25 sin30° = 12.5 m/s

(a) Time of flight: At maximum height, v⊂y; = 0. t⊂up; = u⊂y;/g = 12.5/10 = 1.25 s. Total T = 2 × 1.25 = 2.5 s

(b) Maximum height: H = u⊂y;²/(2g) = 156.25/20 = 7.8 m

(c) Range: R = u⊂x; × T = 21.7 × 2.5 = 54.2 m

Projectile Motion Formula Sheet

Key formulas with definitions, variables, and SI units.
Time of Flight

T = 2u sinθ/g

Total time in the air for a projectile launched at angle θ.

Unit: s (seconds)

Max Height

H = u²sin²θ/(2g)

Maximum height above launch point.

Unit: m (metres)

Range

R = u²sin(2θ)/g

Horizontal distance from launch to landing.

Unit: m (metres)

Horiz. Velocity

u⊂x; = u cosθ

Horizontal component of initial velocity (constant).

Unit: m/s

Vert. Velocity

u⊂y; = u sinθ

Vertical component of initial velocity.

Unit: m/s

SUVAT Vertical

v⊂y; = u⊂y; − gt

Vertical velocity at time t.

Unit: m/s

Frequently Asked Questions

Common questions about this topic, answered clearly.
Projectile motion is the curved path followed by an object launched into the air and moving under gravity alone. The path is parabolic because the horizontal velocity is constant while the vertical velocity changes due to gravitational acceleration.
The horizontal displacement increases linearly with time (constant velocity), while the vertical displacement increases as t squared (constant acceleration). Combining these two gives a parabola: y = x tanθ − gx²/(2u²cos²θ).
The horizontal range R = u²sin(2θ)/g is maximum when sin(2θ) = 1, which occurs when 2θ = 90°, so θ = 45°. At 45°, the projectile achieves the greatest horizontal distance for a given launch speed.
Air resistance opposes motion, reducing the range and maximum height compared to the ideal case. It also changes the optimal angle for maximum range (below 45°) and makes the flight path non-parabolic. For most exam problems, air resistance is ignored.

Related Physics Topics

Mechanics

Full mechanics hub: SUVAT, Newton laws, energy.

Go to Topic →

Newton's Laws

Forces and acceleration — the physics behind projectile motion.

Go to Topic →

Physics Formulas

SUVAT and projectile motion formulas.

Go to Topic →

Scroll to Top