Projectile Motion

Introduction

• Focus on Chapter 1, Lesson 3: Projectile Motion
• Next lesson: Relative Motion

Concept of Projectile Motion in Sports

• Kicking, throwing, or hitting a ball
• Initial contact propels ball upward at an angle
• Ball rises, reaches highest point, then falls due to gravity
• Ignoring air resistance and Earth's rotation, the ball follows a parabolic trajectory
• Horizontal (x) direction: Positive to the right
• Vertical (y) direction: Positive upward
• Ball acts like a projectile affected by gravity
• Example: Ball hit with a tennis racket

Key Points of Projectile Motion

• Horizontal and vertical motions are independent
• Horizontal motion is constant, acceleration = 0
• Vertical motion has constant acceleration due to gravity
• Example: Dropped ball and kicked ball land simultaneously; horizontal range (ΔDx)
• Important property: Horizontal/vertical motions share the same time

Defining Projectile Motion

• Horizontal velocity constant
• Vertical motion has constant acceleration due to gravity
• Horizontal and vertical motions are independent
• Complex projectile motion divided into horizontal uniform motion and vertical uniform acceleration

Analyzing Projectile Motion

• Equations for motion in one dimension applied to X and Y directions separately
• Calculate horizontal (V_ix = V_i * cos(θ)) and vertical (V_iy = V_i * sin(θ)) components of initial velocity
• Positive/negative signs indicate direction

Kinematics Equations

• Horizontal motion (X): Constant velocity
  • V_ix = V_i * cos(θ)
  • ΔDx = V_ix * Δt = V_i * cos(θ) * Δt
• Vertical motion (Y): Constant acceleration (9.8 m/s²)
  • Vfy = V_iy - g * Δt
  • ΔDy = V_iy * Δt - 0.5 * g * Δt²
  • Vfy² = V_iy² - 2g * ΔDy

Sample Problem 1

• Airplane releases supplies (height: 350m, speed: 52 m/s)
  • Calculate time to reach highway and range of package
  • Solution:
    • ΔDy = -350m, V_i = 52 m/s, Δt found using vertical motion equation
    • Range (ΔDx) = 440m

Sample Problem 2

• Golfer hits a golf ball (initial velocity: 25 m/s at 30°)
  • Calculate maximum height and velocity on landing
  • Solution:
    • Maximum height: 8m
    • Velocity on landing: 30.1 m/s at 44° below the horizontal

Range Equation for Projectile Motion

• With zero vertical displacement (ΔDy = 0)
  • Range (ΔDx) = (V_i² / g) * sin(2θ)
  • Angle for maximum range: 45°
  • Air resistance usually ignored

Sample Problem 3

• Soccer ball kicked at 28 m/s at 21°
  • Calculate time in air and distance
  • Solution:
    • Time (Δt): 2.0s
    • Range (ΔDx): 54m

Conclusion

• Review of key concepts and equations
• Homework assignment in Google Classroom