Projectile Motion Equations Cheat Sheet

Key Facts

• Horizontal and vertical motion are independent, so solve $x$ motion and $y$ motion separately using the same time $t$.
• The horizontal velocity component is $v_{0x} = v_0\cos\theta$, and the vertical velocity component is $v_{0y} = v_0\sin\theta$.
• With no air resistance, horizontal acceleration is $a_x = 0$, so horizontal position is $x = x_0 + v_{0x}t$.
• Vertical acceleration near Earth is $a_y = -g = -9.8\,\text{m/s}^2$ when upward is chosen as positive.
• Vertical position is modeled by $y = y_0 + v_{0y}t + \frac{1}{2}a_yt^2$.
• Vertical velocity is modeled by $v_y = v_{0y} + a_yt$, and at maximum height $v_y = 0$.
• For a projectile that lands at the same height it was launched, total flight time is $T = \frac{2v_0\sin\theta}{g}$.
• For a projectile that lands at the same height it was launched, range is $R = \frac{v_0^2\sin 2\theta}{g}$ and maximum height is $H = \frac{v_0^2\sin^2\theta}{2g}$.

Vocabulary

Projectile

A projectile is an object that moves through the air after being launched and is acted on mainly by gravity.

Trajectory

A trajectory is the curved path followed by a projectile during its motion.

Launch Angle

The launch angle is the angle $\theta$ between the initial velocity vector and the horizontal direction.

Initial Velocity Components

Initial velocity components are the horizontal and vertical parts of the launch velocity, given by $v_{0x} = v_0\cos\theta$ and $v_{0y} = v_0\sin\theta$.

Range

Range is the horizontal distance a projectile travels, often represented by $R$.

Maximum Height

Maximum height is the highest vertical position reached by a projectile, where its vertical velocity is $v_y = 0$.