Lesson 2.4 • Competency 2.1Interactive Simulation

Projectile Motion in Two Dimensions

Independence of orthogonal motions, parabolic trajectory equation, time of flight, maximum apex height, and horizontal range on level ground.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

Decompose 2D projectile motion into orthogonal independent horizontal and vertical components.
Calculate time of flight T, apex height H, and horizontal range R analytically and simulate them dynamically.

Core Theory & Concepts

A projectile is any object propelled into free flight subjected only to gravity and initial velocity. Its path is a symmetric parabola formed by the combination of constant horizontal velocity and constant downward vertical acceleration.

Independence of Orthogonal Motions

Horizontal motion proceeds with zero acceleration: a_x = 0 ⇒ v_x = u cos θ (constant). Vertical motion proceeds under gravity: a_y = -g ⇒ v_y = u sin θ - gt.

v_x = u cos θ, v_y = u sin θ - gt

Trajectory Formulas (Level Ground)

• Time of flight: T = (2u sin θ) / g • Maximum height: H = (u² sin² θ) / (2g) • Horizontal range: R = (u² sin 2θ) / g (Max at θ = 45°)

R = (u² sin 2θ) / g, H = (u² sin² θ) / (2g)

Common Misconceptions

Misconception

Thinking a bullet fired horizontally and one dropped from the same height hit the ground at different times.

Physical Clarification:Both bullets have identical initial vertical velocity (u_y = 0) and identical vertical acceleration (a_y = -g). Because vertical motion is completely independent of horizontal motion, both strike the ground simultaneously.

Syllabus Topic Outline

Competency 2.1

1Principle of physical independence of perpendicular motions
2Horizontal motion: constant velocity x = u cos θ · t
3Vertical motion: accelerated motion y = u sin θ · t - ½gt²
4Parabolic trajectory and optimum 45° range condition

Interactive 2D Projectile Motion Laboratory

Simulate projectile kinematics under gravity with real-time vector decomposition, trajectory plotting, and parametric controls.

Scrub Time (s):0.00s

Kinematic Parameters

Initial Velocity u (m s⁻¹):25 m s⁻¹

Launch Angle θ (°):45°

Launch Height h₀ (m):0 m

Gravity g (m s⁻²):9.8 m s⁻²

Real-Time Kinematic Readouts

Time of Flight T:3.61 s

Maximum Height H:15.94 m

Horizontal Range R:63.78 m

Current Height Y:0.00 m

Horizontal Velocity v_x:17.68 m s⁻¹ (constant)

Vertical Velocity v_y:17.68 m s⁻¹

Theoretical Kinematic Principles1. Horizontal Motion: Constant velocity with zero acceleration: x(t) = u·cos(θ)·t 2. Vertical Motion: Constant downward acceleration: y(t) = h₀ + u·sin(θ)·t - 0.5·g·t² 3. Independence of Orthogonal Motions: Horizontal velocity remains invariant (v_x = u·cos(θ)) throughout flight, while vertical velocity varies linearly (v_y = u·sin(θ) - g·t) and reaches zero at the apex.

Practice Problems

A projectile is launched at 20 m s⁻¹ at an angle of 30° to the horizontal. Taking g = 10 m s⁻², calculate the horizontal range in metres.