2D Projectile Motion Overview

Overview

This lecture explains two-dimensional (2D) projectile motion, covering the core equations, examples with both horizontal and angled launches, motion graphs, and how launch angle affects range.

Introduction to 2D Projectile Motion

• 2D projectile motion involves both horizontal (x) and vertical (y) components.
• Gravity only affects the vertical motion (acceleration = 9.8 m/s² downward).
• The horizontal velocity remains constant due to zero horizontal acceleration.
• The initial velocity may be angled, giving both x and y components.
• Time (t) is the same for both x and y motions and connects both sets of equations.

Example 1: Horizontal Launch from a Height

• Object starts at height (e.g., 19.6 m) with initial horizontal velocity (e.g., 10 m/s) and zero initial vertical velocity.
• The x-displacement (range) is determined by the time in air and constant x velocity.
• The y-position decreases with acceleration due to gravity: use y = y₀ + v₀y t + ½ a_y t².
• Time in the air is found when y = 0 (object hits the ground).
• The final speed is found by combining final x and y velocities using the Pythagorean theorem.

Example 2: Launch at an Angle from the Ground

• Object launched with speed v₀ at angle θ (e.g., 22 m/s at 63°).
• Decompose v₀ into v₀x = v₀ cosθ and v₀y = v₀ sinθ.
• X motion: constant x velocity; use x = x₀ + v₀x t.
• Y motion: vertical velocity changes due to gravity; use standard kinematic equations.
• The trajectory is symmetric if initial and final heights are equal.
• At max height, y velocity is zero.
• The final speed when landing is the vector sum of x and y velocity components.

Range and Launch Angle

• Range (Δx) is horizontal displacement and found using Δx = v₀x × total time in air.
• Time in air depends only on vertical motion.
• If initial and final heights are equal: range is maximized at 45° launch angle.
• At angles θ and (90°–θ), the range is the same if initial and final heights are equal.
• For launches from a height, the optimal angle is less than 45°.

Motion Graphs Summary

• X direction: position increases linearly, velocity is constant, acceleration is zero.
• Y direction: position is a curve (parabola), velocity changes linearly, acceleration is constant at -9.8 m/s².
• Graphs differ if initial and final heights are not equal or the launch angle is not upward.

Key Terms & Definitions

• Projectile Motion — motion under gravity with both horizontal and vertical components.
• Range (Δx) — horizontal distance traveled by the projectile.
• Initial Velocity Components — v₀x = v₀ cosθ, v₀y = v₀ sinθ.
• Trajectory — curved path followed by a projectile.
• Maximum Height — highest y position, occurs when vertical velocity is zero.
• Symmetric Trajectory — path where initial and final heights are equal.

Action Items / Next Steps

• Review kinematic equations for x and y directions.
• Practice decomposing velocity vectors into components.
• Solve sample problems for both horizontal and angled launches.
• Memorize the conditions for maximum range and related equations.