Projectile Motion Cheat Sheet

The core ideas of Projectile Motion distilled into a single, scannable reference — perfect for review or quick lookup.

PiqCue — piqcue.com/projectile-motion/cheatsheet

Quick Reference

Independence of Horizontal and Vertical Motion

In projectile motion, the horizontal and vertical components of motion are independent. Horizontal velocity remains constant (no air resistance), while vertical velocity changes due to gravitational acceleration.

Trajectory

The curved path a projectile follows through space, which forms a parabola under uniform gravity with no air resistance. The shape depends on launch angle, speed, and gravitational acceleration.

Range

The horizontal distance a projectile travels from launch to landing. For a projectile launched from and landing at the same height, range is R = (v0^2 sin 2theta) / g.

Time of Flight

The total time a projectile spends in the air from launch to landing. It depends on the vertical component of the initial velocity and the height difference between launch and landing points.

Launch Angle

The angle between the initial velocity vector and the horizontal. It determines how the initial speed is split between horizontal and vertical components, directly affecting the shape and range of the trajectory.

Maximum Height

The highest vertical position reached by a projectile, occurring when the vertical component of velocity momentarily equals zero. The formula H = v_0^2 sin^2(theta)/(2g) shows maximum height depends on the square of the vertical velocity component. A steeper launch angle yields greater height for the same initial speed.

Free Fall and Gravitational Acceleration

Free fall is motion under the sole influence of gravity, with acceleration g (approximately 9.8 m/s^2) directed downward near Earth surface. In projectile motion, the vertical component is always in free fall regardless of horizontal motion. Gravitational acceleration is constant near the surface.

Reference Frames in Projectile Motion

A reference frame is the coordinate system and state of motion from which observations are made. The trajectory of a projectile can look very different depending on the observer reference frame. Choosing a convenient frame simplifies the mathematics dramatically.

Key Terms at a Glance

Projectile:An object launched into the air moving under gravity alone, with no propulsion during flight.

Trajectory:The curved path traced by a projectile; a parabola under ideal conditions with no air resistance.

Launch angle:The angle between the initial velocity vector and the horizontal at the moment of launch.

Range:The horizontal distance from launch to landing at the same elevation.

Time of flight:The total time a projectile remains airborne from launch to landing.

Maximum height:The highest vertical position reached, where vertical velocity momentarily equals zero.

Free fall:Motion under gravity alone, with acceleration g (approximately 9.8 m/s^2) downward.

Kinematic equations:Equations relating displacement, velocity, acceleration, and time for uniformly accelerated motion.

Velocity components:The horizontal and vertical parts of a velocity vector, resolved along perpendicular axes.

Air resistance (drag):A force opposing motion through air, dependent on speed, cross-sectional area, and air density.

Complementary angles:Two angles summing to 90 degrees; they produce equal ranges on level ground without air resistance.

Parabola:A symmetric curve described by a quadratic equation, representing an ideal projectile trajectory.

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