Lecture on Projectile Motion

Key Concepts

• Projectile Launched from Height: The projectile is launched from a height at an angle ( \theta ).
• Objective: Calculate various parameters such as horizontal range, time in air, horizontal and vertical velocities, and the impact angle.

Given Information

• Initial Velocity ( v_0 ): 10 m/s
• Angle ( \theta ): 45 degrees
• Height: 2 meters

Calculating Initial Velocity Components

• Horizontal Component: ( v_{0x} = v_0 \cos(\theta) = 10 \cos(45^\circ) = 7.07 \text{ m/s} )
• Vertical Component: ( v_{0y} = v_0 \sin(\theta) = 10 \sin(45^\circ) = 7.07 \text{ m/s} )
  • Note: At 45 degrees, both components are equal.

Time of Flight

• Use the kinematic equation: ( y = v_{0y}t - \frac{1}{2}gt^2 )
• Final Height ( y ): 0 m
• Initial Height ( y_0 ): 2 m
• Solve the quadratic equation for time:
  • Two solutions: 1.69 s and 0.24 s
  • Feasible Time: 1.69 s (negative time values are ignored for practical scenarios)

Horizontal Distance

• Equation: ( x = v_{0x}t + \frac{1}{2}a_xt^2 )
  • Acceleration ( a_x ): 0 (No horizontal acceleration)
• Horizontal Range: 11.925 meters

Impact Velocities

• Horizontal Velocity at Impact: Remains constant at 7.07 m/s
• Vertical Velocity at Impact:
  • Use: ( v_{gy} = v_{0y} - gt )
  • Final velocity ( v_{gy} = -9.49 \text{ m/s} )
  • Negative sign indicates downward direction

Impact Angle

• Calculate using tangent function: ( \tan(\alpha) = \frac{v_{gy}}{v_{gx}} )
  • ( \alpha = \tan^{-1}\left(\frac{-9.49}{7.07}\right) = -53.3^\circ )
  • Negative direction indicates downward trajectory post-impact

Magnitude of Impact Velocity

• Using Pythagorean theorem: ( v_{impact} = \sqrt{v_{gx}^2 + v_{gy}^2} )
  • ( v_{impact} = 11.83 \text{ m/s} )

Conclusion

• Exercise: Re-calculate the problem for a 30-degree launch angle.
• Recommendation: Review and practice similar problems for better understanding.
• Engagement: Encourage subscribing for more lectures.

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This summary captures the critical calculations and inferences necessary for understanding projectile motion from a given height and angle.