Overview
This lecture reviews essential kinematic equations and explains how to solve common projectile motion problems involving different types of trajectories.
Basic Kinematic Equations
- Displacement for constant velocity: ( d = vt )
- Final velocity with constant acceleration: ( v_f = v_i + at )
- Displacement using average velocity: ( d = \frac{v_i + v_f}{2}t )
- Displacement using initial velocity and acceleration: ( d = v_it + \frac{1}{2}at^2 )
- Velocity-squared equation: ( v_f^2 = v_i^2 + 2ad )
- Displacement (( d )) means the difference between final and initial positions, can apply in ( x ) or ( y ) directions.
Horizontal Launch from a Height (Type 1)
- Height: ( h = \frac{1}{2}gt^2 ) (uses vertical motion only, ( v_{iy} = 0 ))
- Range: ( R = v_{x}t ) (( v_x ) is constant; ( a_x = 0 ))
- To find speed before impact: ( v = \sqrt{v_x^2 + v_y^2} ) where ( v_{yf} = gt )
- Angle before hitting ground: ( \theta = \arctan \left( \frac{v_y}{v_x} \right) )_
Angled Launch from Ground (Type 2)
- Initial velocity components: ( v_x = v \cos \theta ), ( v_y = v \sin \theta )
- Time to max height: ( t_{up} = \frac{v \sin \theta}{g} )
- Total flight time: ( t_{total} = \frac{2v \sin \theta}{g} )
- Maximum height: ( h_{max} = \frac{v^2 \sin^2 \theta}{2g} )
- Range: ( R = \frac{v^2 \sin 2\theta}{g} )
- At landing, speed and angle match launch values (symmetrical path)._
Angled Launch from a Height (Type 3)
- Use: ( y_f = y_i + v_{iy}t + \frac{1}{2} a t^2 ) (for vertical position)
- Set ( y_f = 0 ) to solve for time ( t ) (may require quadratic formula).
- Alternative: Find time up then time down, sum for total time.
- Range: ( R = v_x t_{total} ) where ( v_x = v \cos \theta )
- Do not use the symmetrical range formula unless the trajectory is symmetrical.
Calculating Final Speed and Angle Upon Impact
- ( v_x ) is constant throughout the motion.
- Final vertical velocity: ( v_{yf} = v_{iy} - gt )
- Impact speed: ( v = \sqrt{v_x^2 + v_{yf}^2} )
- Impact angle: ( \theta = \arctan \left( \frac{|v_{yf}|}{v_x} \right) )
- Describe angle as degrees below the horizontal or relative to the positive ( x )-axis as appropriate.
Key Terms & Definitions
- Displacement — The straight-line distance and direction from initial to final position.
- Range (( R )) — Horizontal distance traveled by the projectile.
- Maximum Height (( h_{max} )) — Maximum vertical displacement above launch point.
- Projectile Motion — Motion involving both horizontal and vertical components under constant acceleration due to gravity._
Action Items / Next Steps
- Memorize the core kinematic and projectile motion equations.
- Practice distinguishing which equation to use for each type of trajectory.
- Complete assigned projectile motion practice problems, focusing on separating ( x ) and ( y ) components.