Lecture on Projectile Motion

Overview

• Discussed a projectile thrown from the ground.
• Initial angle ( \theta ) is 30 degrees.
• Initial velocity ( V_0 ) is 25 m/s.

Objectives

• Determine time to reach maximum height.
• Calculate horizontal and vertical velocities at the peak.
• Find time for projectile to return to the ground.
• Determine horizontal range.
• Find position at ( t = 2.5 ) seconds.

Key Concepts

• Horizontal Velocity ( V_{0x} = V_0 \cos(\theta) )
  • ( V_{0x} = 21.65 ) m/s
• Vertical Velocity ( V_{0y} = V_0 \sin(\theta) )
  • ( V_{0y} = 12.5 ) m/s
• Acceleration
  • ( a_x = 0 )
  • ( a_y = -9.8 ) m/s² (gravity)

Time to Reach Maximum Height

• Equation: ( V_y = V_{0y} - gt )
• At the peak: ( V_y = 0 )
• Solving for time ( t ):
  • ( t = \frac{V_{0y}}{g} = \frac{12.5}{9.8} = 1.28 ) seconds

Maximum Height

• Equation: ( V_y^2 = V_{0y}^2 - 2gH )
• Solving for ( H ):
  • ( H = \frac{V_{0y}^2}{2g} = \frac{12.5^2}{2 \times 9.8} = 7.97 ) meters

Velocities at the Peak

• Horizontal Velocity: Remains constant at 21.65 m/s
• Vertical Velocity: 0 m/s at the peak

Total Time of Flight

• Time to reach ground is twice the time to reach peak:
  • ( 2 \times 1.28 = 2.56 ) seconds

Velocities upon Hitting the Ground

• Horizontal Velocity: 21.65 m/s
• Vertical Velocity: -12.5 m/s (symmetrical to launch)
• Impact Angle: Same as launch angle (30 degrees)

Horizontal Range

• Equation: ( R = V_{0x} \times 2 \times t )
  • ( R = 21.65 \times 2 \times 1.28 = 55.42 ) meters_

Position at ( t = 2.5 ) seconds

• X Position: ( x = V_{0x} \times t = 21.65 \times 0.5 = 10.83 ) meters
• Y Position: ( y = V_{0y} \times t - \frac{1}{2} g t^2 )
  • ( y = 12.5 \times 0.5 - \frac{1}{2} \times 9.8 \times (0.5)^2 = 5.025 ) meters

Conclusion

• Solve problems by breaking into x and y components.
• Avoid memorizing formulas; rely on understanding kinematic equations.
• Practice with different angles, e.g., 60 degrees.
• Recommendation: Solve similar problems for better understanding.

Reminder

• Like, share, and subscribe for more content.