Overview
This lecture covers the translation of kinematic equations to vertical (y-axis) motion, focusing on free-falling and upward-thrown objects, and demonstrates problem-solving using these equations.
Kinematic Equations (Review and Conversion)
- The four basic kinematic equations relate velocity, position, acceleration, and time for uniformly accelerated motion.
- For y-axis motion: replace x with y, ax with ay, and vx with vy.
- The standard acceleration in y (gravity) is ay = -g, where g = 9.8 m/s² (positive constant).
Kinematic Equations for y-Dimension
- Final velocity: vf = vi - g·t
- Position: yf = yi + ½(vi + vf)·t
- Position (expanded): yf = yi + vi·t - ½g·t²
- Velocity-squared: vf² = vi² - 2g(yf - yi)
- "Initial" and "final" can refer to any two chosen points in the object's motion.
Example Problem: Stone Thrown Upward from a Building
- Initial velocity (vi or VA) = 20 m/s upward from roof (yi = 0).
- Building height H = 50 m.
Part A: Time to Maximum Height
- At maximum height, velocity vf (VB) = 0.
- Use vf = vi - g·t: solve for t_max = vi/g = 20/9.8 ≈ 2.04 s.
Part B: Maximum Height Achieved
- Use yf = yi + ½(vi + vf)·t or yf = yi + vi·t - ½g·t² with t = t_max.
- Maximum height above roof: yf ≈ 20.4 m.
- Alternatively, use vf² = vi² - 2g(yf - yi) without finding the time.
Part C: Velocity Upon Return to Roof
- On return (downward), velocity magnitude equals initial but direction is negative: vf = -20 m/s.
- Methods include time-based equations or conservation of energy/symmetry.
Part D: Velocity and Position at t = 5 s
- Velocity: vf = vi - g·t = 20 - 9.8·5 = -29 m/s (downward).
- Position: yf = yi + vi·t - ½g·t² = 0 + 20·5 - 0.5·9.8·25 = -22.5 m (below the roof).
Interpreting Signs and Magnitude
- Negative velocity or position means downward or below the reference point (roof).
- Speed is the magnitude of velocity (ignore the sign).
Key Terms & Definitions
- Kinematic Equations — Formulas for motion with constant acceleration.
- g — Acceleration due to gravity, 9.8 m/s² down.
- Free Fall — Motion under gravity alone, ay = -g.
- Initial/Final — Selected start/end points in the motion (not necessarily the absolute start/end).
- Speed — Magnitude of velocity (always positive).
- Velocity — Vector quantity, can be positive (up) or negative (down).
Action Items / Next Steps
- Write down and memorize the four y-dimension kinematic equations for exams.
- Complete homework problems on free-fall and vertical motion.
- No need to study Section 9 (calculus-based derivations) for the test.