Lecture on Projectile Motion

Basics of Kinematics

• Kinematic Equations:
  • At constant speed: ( d = vt ) (Displacement = velocity ( \times ) time)
  • Under constant acceleration:
    • Displacement = average velocity ( \times ) time
    • Average velocity = ((V_{initial} + V_{final}) / 2)
  • Equations:
    • ( V_{final} = V_{initial} + at )
    • ( V_{final}^2 = V_{initial}^2 + 2a \times d )
    • ( d = V_{initial} \times t + \frac{1}{2} a t^2 )
  • Displacement could be along x-axis or y-axis
  • Distance and displacement are the same in one direction; different if direction changes_

Example: Kicking a Ball

• Gravitational acceleration: (-9.8 \text{ m/s}^2) (approx. (-10) for simplicity)
• Initial Conditions:
  • Horizontal velocity ((Vx)): 7 \text{ m/s}
  • Vertical velocity ((Vy)): 30 \text{ m/s}
• Velocity Changes Over Time:
  • (Vx) remains constant (7 \text{ m/s})
  • (Vy) decreases by 10 every second due to gravity
  • At the peak (Vy = 0): only horizontal motion
  • Post-peak: (Vy) becomes negative
• Speed vs Velocity:
  • Speed is the magnitude of velocity (always positive)
  • Velocity is vector (can be negative)

Types of Projectile Trajectories

1. Horizontal Launch from a Cliff:
   • Height equation: ( h = \frac{1}{2} a t^2 )
   • Range: ( R = Vx \times t )
2. Projectile Launched at an Angle from Ground:
   • Time to peak: ( t = \frac{V \sin \theta}{g} )
   • Total time: twice the time to peak
   • Maximum Height: ( h_{max} = \frac{V^2 \sin^2 \theta}{2g} )
   • Range: ( R = \frac{V^2 \sin 2\theta}{g} )
3. Projectile Launched at an Angle from a Height:
   • Uses previous equations plus:
   • Quadratic formula for time to ground_

Additional Concepts

• Velocity Components:
  • ( Vx = V \cos \theta )
  • ( Vy = V \sin \theta )
• Projectile Definitions:
  • Only under influence of gravity; air resistance ignored

Sample Problems

1. Horizontal Launch: Given initial horizontal velocity, time to ground;
   • Calculate height and range
   • Equation: ( h = \frac{1}{2} a t^2 )
2. Vertical Drop with Initial Speed:
   • Quadratic formula utilized to solve for time with initial vertical speed
   • Adjust equations for the scenario including initial velocity impact on timing

Equations Summary

• Quadratic Formula: For complex time calculations
  • ( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
• Pythagorean Relation: ( V^2 = Vx^2 + Vy^2 )
• Angle Calculation: ( \theta = \tan^{-1}(\frac{Vy}{Vx}) )