Overview
This lecture covers the fundamentals of projectile motion, including key equations, coordinate system choices, and example problems relevant to two-dimensional kinematics.
What is Projectile Motion?
- A projectile is an object launched and moving under the influence of gravity only, with no propulsion.
- Projectile motion problems ignore air resistance for simplicity.
- The motion can be treated as two-dimensional by aligning axes with the trajectory plane.
Choosing a Coordinate System
- The vertical axis (+y) is chosen perpendicular to the ground, with gravity acting downward (negative y direction).
- The horizontal axis (+x) is parallel to the ground, with zero acceleration due to the neglect of air resistance.
Kinematics of Projectile Motion
- The general kinematic equations for constant acceleration apply separately to the x and y directions.
- Acceleration in x-direction: ( a_x = 0 ); in y-direction: ( a_y = -g ) (gravity).
- Initial velocity components: ( v_{0x} = v_0 \cos \theta ), ( v_{0y} = v_0 \sin \theta ).
- Position as a function of time:
- ( x(t) = v_0 \cos\theta \cdot t )
- ( y(t) = v_0 \sin\theta \cdot t - \frac{1}{2}gt^2 )
Maximum Height and Range
- At maximum height, vertical velocity equals zero.
- Time to max height: ( t_{top} = \frac{v_0 \sin\theta}{g} ).
- Maximum height: ( y_{max} = \frac{(v_0 \sin\theta)^2}{2g} ).
- Total time in air (if landing at launch height): ( t_{flight} = \frac{2v_0 \sin\theta}{g} ).
- Range (horizontal distance): ( R = \frac{v_0^2 \sin2\theta}{g} ).
Important Notes on Range
- The range equation applies only if the projectile lands at the same height as launch.
- Two launch angles can yield the same range for a given initial speed.
- Increasing gravity decreases range, time in the air, and height.
Sample & Practice Problems
- Problems include launching from slopes, over cliffs, at different heights, and adjusting for ceiling height.
- Example: Calculating where a rock lands after being thrown from a cliff or optimizing launch angle to just graze the ceiling.
Key Terms & Definitions
- Projectile — An object moving under the influence of gravity only after launch.
- Kinematics — The study of motion without regard to forces.
- Initial velocity (( v_0 )) — Speed and direction at the moment of launch.
- Range (( R )) — The horizontal distance traveled by a projectile.
- Gravity (( g )) — Acceleration due to Earth's gravity, approximately ( 9.8, \text{m/s}^2 ).
Action Items / Next Steps
- Solve the listed practice problems involving angled launches, non-level ground, and varying initial heights.
- Review the kinematic equations for both x and y directions.
- Use the provided equations to analyze different projectile scenarios.