Mechanics · Quick Reference

Projectile Motion Cheat Sheet

Mechanics · Lesson 6/19

In one line: a projectile is any object launched into the air and left to move under gravity alone — its path traces a graceful curve called a parabola.

Key Ideas

1Projectile Motion Setup. Combines constant horizontal velocity (no horizontal acceleration) with constant vertical acceleration due to gravity (g ≈ 9.8 m/s^2 downward).

2Horizontal Motion. Since there's no horizontal force (ignoring air resistance), horizontal velocity stays constant throughout the flight: x = vx*t.

3Vertical Motion. Vertical velocity changes due to gravity: vy = vy0 - g*t, and vertical position follows y = vy0*t - (1/2)*g*t^2.

4Time of Flight. For a projectile launched and landing at the same height, time of flight = 2*vy0/g.

5Maximum Height and Range. Maximum height = vy0^2/(2g). Range (horizontal distance) = vx*(time of flight), often written as R = v^2*sin(2*theta)/g for launch speed v and angle theta.

Worked Examples

A ball is thrown horizontally at 10 m/s from a height where it takes 2 seconds to hit the ground. Find the horizontal distance traveled.

20 meters

A projectile is launched straight up at 20 m/s (g=10 m/s^2). Find the time to reach maximum height.

2 seconds

A projectile is launched at 30 m/s at a 45° angle (g=10 m/s^2, sin(90°)=1). Find its range.

90 meters

Formulas

Time of flight (same launch/landing height) = 2*vy0/g

Maximum height = vy0^2 / (2g)

Range = v^2 * sin(2*theta) / g

Practice Yourself

A ball is thrown horizontally at 5 m/s and falls for 3s. Find horizontal distance.

15 m

A projectile is launched straight up at 30 m/s (g=10). Find time to max height.

Find the max height for vy0=20 m/s (g=10).

20 m (using vy0^2/(2g)=400/20)