Lecture Notes on Projectile Motion

Introduction

• Discussion on equations necessary for solving projectile motion problems.
• Review of basic kinematic equations.

Basic Kinematic Equations

1. Constant Speed:

   • Displacement = Velocity x Time

2. Constant Acceleration:

   • Equation 1: Final speed = Initial speed + (Acceleration x Time)
   • Equation 2: Final speed² = Initial speed² + 2 x (Acceleration x Displacement)
   • Equation 3: Displacement = Average speed x Time
     • Average speed = (Initial speed + Final speed) / 2
   • Equation 4: Displacement = Initial velocity x Time + 1/2 x Acceleration x Time²

Concepts of Displacement and Distance

• Displacement (D) can be used interchangeably with distance when the object moves in one direction.
• Formula for displacement: Final position - Initial position
• Displacement can also be considered in x or y directions.

Projectile Motion Types

Type 1: Horizontal Projectile

• Object falls off a cliff horizontally.

Key Equations:

1. Height Equation:
   • h = 1/2 x a x t²
   • Applied in the y-direction: dy = vy_initial x t + 1/2 x ay x t²
2. Range Equation:
   • Range = vx x t

Notes:

• Vx remains constant (acceleration in the x-direction is 0).
• Vy changes due to gravitational acceleration (-9.8 m/s²).
• Speed before hitting the ground can be found using both horizontal and vertical velocities.
• Calculate angle using: inverse tangent (Vy/Vx)

Type 2: Launched from Ground

• Object launched at angle, forming an arc.

Key Equations:

1. Time to Max Height (A to B):
   • t = v sin(theta) / g
2. Total Time (A to C):
   • t = 2v sin(theta) / g
3. Maximum Height:
   • h = v² sin²(theta) / 2g
4. Range:
   • Range = v² sin(2theta) / g

Notes:

• Symmetrical graph implies speed and angle at A are the same at C.

Type 3: Launched from a Cliff

• Object launched from an elevated position.

Key Equations:

1. Time to Hit Ground (A to C):
   • Use quadratic formula for time:
     • Displacement in y: y_final = y_initial + vy_initial x t + 1/2 x g x t²
2. Range:
   • Range = vx x total time

Alternative Method for Time:

• Calculate time to max height and use total height for time from B to C.
• Total height = h (cliff) + h (max height from launch)

Additional Considerations

• Vx is constant across all points (A, B, C).
• Use inverse tangent for angle relative to horizontal.
• For symmetrical trajectories, speed and angle at launch equal to those before hitting ground.

Conclusion

• Review of all equations needed for projectile motion problems.