Overview
The lecture reviews projectile motion, emphasizing the separation of horizontal and vertical components, key kinematic equations, and special cases like time of flight, trajectory, and range.
Independence of X and Y Motion
- Projectile motion separates into independent horizontal (x) and vertical (y) motions.
- Horizontal (x) motion has constant velocity: (v_x = v \cos\theta).
- Vertical (y) motion is free fall with constant acceleration: (a_y = -g), and (v_y = v \sin\theta).
Kinematic Equations for Projectile Motion
- Generic kinematic equations: (v = v_0 + at), (x = x_0 + v_0 t + \frac{1}{2} a t^2), (v^2 = v_0^2 + 2a(x - x_0)).
- For x-direction: (a_x = 0), so (v_x = v_{0x}) and (x = x_0 + v_{0x} t).
- For y-direction: (a_y = -g), so (v_y = v_{0y} - gt), (y = y_0 + v_{0y} t - \frac{1}{2} g t^2).
Application Examples
- When an object is dropped midair, it stays directly beneath its moving source, maintaining horizontal velocity.
- Time to hit the ground is (t = \sqrt{\frac{2 \Delta y}{g}}).
- Packages dropped from a moving plane and a stationary balloon at the same height land simultaneously.
Special Projectile Scenarios
Time of Flight (TOF) for Flat Surfaces
- Valid only if launch and landing heights are equal ((y_{\text{launch}} = y_{\text{impact}})).
- Time of flight: (t_{TOF} = \frac{2 v_0 \sin\theta}{g}).
Trajectory Equation (Path)
- Eliminating time gives: (y = x \tan\theta - \frac{g}{2 v_0^2 \cos^2\theta} x^2).
- This represents a parabolic path.
Range Formula
- Range (R = \frac{v_0^2 \sin 2\theta}{g}).
- Applies only when projectile lands at same height as launched.
Key Terms & Definitions
- Projectile Motion — Motion under gravity with initial velocity at an angle.
- Time of Flight — Total time the projectile is in the air.
- Trajectory — The path (y as a function of x) of the projectile.
- Range — Horizontal distance traveled by the projectile.
Action Items / Next Steps
- Practice rearranging kinematic equations for given problems.
- Reflect on how changes in gravity affect time of flight and range.
- Solve example problems using the provided formulas.