Overview
This lecture covers projectile motion as a special case of general motion under gravity, focusing on key principles, equations, and solving real-world examples involving horizontal and inclined launches.
Projectile Motion Fundamentals
- Projectile motion involves objects moving under Earth's gravity alone.
- The horizontal velocity remains constant since there is no horizontal acceleration.
- The vertical velocity changes continuously due to constant acceleration from gravity (g = 9.81 m/s²).
- The trajectory of a projectile forms a parabolic path.
Equations of Motion for Projectiles
- Horizontal position: ( x = x_0 + v_{0x} t ) (since ( a_x = 0 ))
- Vertical position: ( y = y_0 + v_{0y} t - \frac{1}{2} g t^2 )
- Vertical velocity: ( v_y = v_{0y} - g t )
- Vertical velocity squared: ( v_y^2 = v_{0y}^2 - 2g(y - y_0) )
Example 1: Soccer Ball Kick
- Ball kicked at ( 10 ) m/s, ( 30^\circ ) above ground.
- ( v_{0x} = 10 \cos 30^\circ ), ( v_{0y} = 10 \sin 30^\circ ).
- Time to hit ground found by solving ( y = y_0 ): ( t = 1.01 ) s.
- Horizontal distance: ( x = v_{0x} t = 8.83 ) m.
Example 2: Cannon on Cliff
- Cannon fired at ( 150 ) m/s, incline with rise/run ( 3/4 ).
- ( v_{0x} = (4/5)\times150 ), ( v_{0y} = (3/5)\times150 ).
- Solve quadratic for time when projectile lands: ( t = 19.89 ) s.
- Range: ( x = v_{0x} t = 2387 ) m.
Example 3: Golf Ball on Inclined Plane
- Ball hits at ( 80 ) ft/s, ( 45^\circ ) to ground; ground inclines at ( 10^\circ ).
- Use reference frame where ( y ) is vertical, ( x ) is horizontal.
- ( v_{0x} = 80\cos(55^\circ) ), ( v_{0y} = 80\sin(55^\circ) ).
- Distance calculation involves both ( x ) and ( y ) components: ( D = \sqrt{ x^2 + y^2 } ).
- Distance before first bounce: ( D = 166 ) ft.
Key Terms & Definitions
- Projectile motion — motion under constant acceleration due to gravity, ignoring air resistance.
- Horizontal velocity (( v_{0x} )) — initial velocity component parallel to ground, remains constant.
- Vertical velocity (( v_{0y} )) — initial velocity component perpendicular to ground, changes due to gravity.
- g — acceleration due to gravity (( 9.81 ) m/s² or ( 32.2 ) ft/s²).
Action Items / Next Steps
- Practice solving projectile motion problems using the provided equations.
- Review kinematic equations for constant acceleration.
- Prepare for homework on projectile motion scenarios, including inclined planes.