Lecture Notes on Projectile Motion

Introduction

• Focus on applying previously learned concepts rather than introducing new material.
• Discuss trajectory of a projectile (e.g., golf ball or tennis ball) shot at an angle.

Components of Projectile Motion

• Initial Conditions:
  • Horizontal component:
    • ( v_{0x} = v_0 \cos{\alpha} )
  • Vertical component:
    • ( v_{0y} = v_0 \sin{\alpha} )
• The motion can be analyzed using the equations of motion.
• Constants needed: ( x_0, v_{0x}, v_{0y} ) chosen as zero arbitrarily.

Motion Analysis

• Vertical Motion:
  • Changes due to acceleration (gravity): ( a = -g ) (with ( g \approx 9.8 ) m/s²)
  • Equation for vertical position:
    • ( y(t) = v_{0y} t - \frac{1}{2} g t^2 )
• Horizontal Motion:
  • Constant velocity: ( x(t) = v_{0x} t )

Shape of the Trajectory

• By eliminating time, the trajectory can be expressed as a function of ( x ):
  • Resulting in a parabolic equation:
    • ( y = C_1 x - C_2 x^2 ) (where ( C_1, C_2 ) are constants)

Maximum Height Calculation

• To find the time to reach maximum height (point P):
  • Set vertical velocity to 0:
    • ( t_P = \frac{v_{0y}}{g} = \frac{v_0 \sin{\alpha}}{g} )
• Maximum height above ground:
  • ( h = \frac{v_{0y}^2}{2g} = \frac{(v_0 \sin{\alpha})^2}{2g} )

Total Time of Flight

• Total time to reach the ground (point S):
  • Two times the time to reach maximum height:
    • ( t_S = 2 \cdot t_P = \frac{2 v_0 \sin{\alpha}}{g} )

Horizontal Range Calculation

• Range (distance OS) when landing:
  • ( OS = \frac{v_0^2 \sin{2\alpha}}{g} )

Key Observations

• Higher initial speed or angle results in higher maximum height.
• The horizontal range is influenced by the initial speed squared.
• Optimal angle for maximum range is 45 degrees.

Experimental Setup

• Shooting a ball at various angles (30°, 45°, 60°) and predicting landing spots.
• Importance of measuring uncertainties in the measurements.

Uncertainties in Measurements

• Measurement of maximum height leads to calculation of v0 squared.
• Height measurement example:
  • If height is 3.07 m, then ( v_0^2 = 60.2 ) m²/s².
• Angle measurement has to be accurate, especially for angles other than 45°.

Monkey and the Hunter Analogy

• Discusses the trajectory of a projectile aimed at a moving target (the monkey) and how gravity affects both the projectile and the monkey.
• Analysis from both the hunter's and monkey's perspective illustrates projectile motion concepts.

Conclusion

• Emphasizes the importance of understanding projectile motion and its real-world applications and experiments.
• Highlights the need to account for uncertainties in experimental physics.