Projectile Motion and Kinematic Equations
Basic Concepts
- Displacement with Constant Speed:
- Constant Acceleration Equations:
- ( V_f = V_i + at )
- ( V_f^2 = V_i^2 + 2aD )
- ( D = V_{avg} \times T )
- ( V_{avg} = \frac{V_i + V_f}{2} )
- ( D = V_i \times T + \frac{1}{2} a T^2 )
Displacement and Distance
- Displacement (D): Difference between final and initial position.
- Can be in X or Y direction.
Types of Projectile Trajectories
Type 1: Horizontal Launch from a Cliff
- Vertical Motion:
- ( H = \frac{1}{2} a_y T^2 )
- At the top, ( V_{y, initial} = 0 ), so ( H = \frac{1}{2} g T^2 )
- Horizontal Motion:
- ( Range = V_x T )
- ( V_x ) is constant, as ( a_x = 0 )
- Vertical Final Velocity: ( V_{y, final} = V_{y, initial} + aT )
- Final Speed Before Impact: Combine ( V_x ) and ( V_{y, final} )
- Angle: Use ( \tan^{-1} \left( \frac{V_y}{V_x} \right) )
Type 2: Launch at an Angle from Ground
- Components of Initial Velocity:
- ( V_x = V \cos(\theta) )
- ( V_y = V \sin(\theta) )
- Time to Reach Maximum Height (A to B):
- ( T_{A \to B} = \frac{V \sin(\theta)}{g} )
- Total Time of Flight (A to C):
- ( T_{A \to C} = 2 \times \frac{V \sin(\theta)}{g} )
- Maximum Height:
- ( H = \frac{V^2 \sin^2(\theta)}{2g} )
- Range:
- ( R = \frac{V^2 \sin(2\theta)}{g} )
Type 3: Launch at an Angle from a Height
- Vertical Displacement:
- ( y_{final} = y_{initial} + V_{y, initial} T + \frac{1}{2} g T^2 )
- Use quadratic formula to find ( T )
- Alternative Time Calculation: Sum of times from A to B and B to C
- Range:
- Final Speed Before Impact:
- Calculate ( V_{y, final} ) using ( V_{y, final} = V_{y, initial} - gT )
Additional Considerations
- Constant Horizontal Velocity: ( V_x ) remains unchanged during flight.
- Angle Calculations: Ensure the correct reference angle relative to your problem context (e.g., below horizontal vs. relative to positive x-axis).
These equations and their application to different projectile motion scenarios are crucial for problem-solving in physics.