Lecture Notes on Projectile Motion

Introduction

• Emphasis on applying learned concepts rather than introducing new ones.
• Focus on understanding the trajectory of projectiles, such as golf and tennis balls.

Trajectory of Projectile

• Projectile shot at angle ( \alpha )
• Horizontal component: ( v_{0x} = v_0 \cos(\alpha) )
• Vertical component: ( v_{0y} = v_0 \sin(\alpha) )
• Points of interest: Highest point ( P ) and ground impact point ( S ).

Equations of Motion

• Use of one-dimensional equations:
  • No acceleration in x-direction.
  • Acceleration in y-direction: ( a_y = -g = -9.8 , m/s^2 )
• Selected constants: ( x_0 = 0 ), ( y_0 = 0 )

Vertical Motion Equation

• ( y(t) = v_{0y}t - \frac{1}{2}gt^2 )
• Substitute ( v_{0y} ): ( y(t) = (v_0 \sin(\alpha))t - \frac{1}{2}gt^2 )

Horizontal Motion Equation

• ( x(t) = v_{0x}t )
• Substitute ( v_{0x} ): ( x(t) = (v_0 \cos(\alpha))t )
• Eliminating time ( t ) gives the relationship between ( y ) and ( x ).

Parabola Shape of Trajectory

• Resulting equation: ( y = Cx - Kx^2 )
• Confirms shape of trajectory is a parabola (second-order equation).

Time to Reach Highest Point

• Highest point reached when vertical velocity is zero:
  • ( v_{0y} - gt = 0 ) leads to ( t_P = \frac{v_0 \sin(\alpha)}{g} )

Highest Point Calculation

• Substitute ( t_P ) back into the vertical equation to find the maximum height ( h ):
  • ( h = \frac{(v_0 \sin(\alpha))^2}{2g} )

Time to Reach Point S

• Time to hit the ground ( t_S = 2t_P = \frac{2v_0 \sin(\alpha)}{g} )
• This is based on symmetry of the projectile motion.

Distance Traveled (Range)

• Range ( OS = v_{0x} \times t_S )
• Substitute to find:
  • ( OS = \frac{v_0^2 \sin(2\alpha)}{g} )
• This expression shows the dependence on ( v_0^2 ) and ( \sin(2\alpha) ).

Importance of Uncertainties

• Measurement uncertainties must be acknowledged in experiments.
• Example: Measuring the height of a projectile to determine initial velocity and its uncertainties.

Angle of Projection Effects

• Projectiles launched at different angles (30°, 45°, 60°):
  • Predicts hit point on a target based on calculated distances.
  • Higher angles provide different trajectories and distances.

Practical Application of Predictions

• Experimental verification of trajectory calculations with actual projectile launches.
• Use of a metal ball and spring gun to demonstrate principles.

Tragic Scenario: Monkey and Hunter

• A hunter shoots at a monkey; the monkey drops when the gun fires.
• The trajectory of the bullet and the falling monkey are analyzed:
  • The bullet follows a parabolic path while the monkey falls straight down.
• Elaborates on the independence of speed and angle in this scenario.

Perspective from the Monkey

• The monkey’s point of view is analyzed in a falling elevator scenario.
• Both the bullet and the monkey fall together, creating a straight line situation.

Conclusion

• The lecture concludes with an experimental demonstration involving the monkey and the bullet's trajectory, emphasizing the calculations and uncertainties involved in projectile motion.