Overview
This lecture covers problem-solving techniques for kinematics in one dimension, focusing on constant velocity and uniform (constant) acceleration scenarios, with step-by-step examples and the main equations used.
Kinematics Scenarios
- Kinematics deals with motion in one dimension involving displacement, velocity, and acceleration.
- Two main scenarios: constant velocity (no acceleration) and uniform (constant) acceleration.
- Algebra-based physics courses focus only on constant and uniform acceleration, not varying acceleration.
- Each scenario has a small set of equations for solving problems.
Equations for Constant Velocity
- For no acceleration (constant velocity):
- Displacement ((\Delta x)) = velocity ((v)) × time ((t)), or (\Delta x = vt).
Equations for Uniform (Constant) Acceleration
- Average velocity ((v_{avg})) = (initial velocity (v_0) + final velocity (v)) / 2.
- Displacement: (\Delta x = v_{avg} t = \frac{v_0 + v}{2} t).
- Displacement: (\Delta x = v_0 t + \frac{1}{2} a t^2), where (a) is acceleration.
- Velocity: (v = v_0 + at).
- (v^2 = v_0^2 + 2a\Delta x) (used when time is not known).
Systematic Problem-Solving Approach
- Identify whether the problem involves constant velocity or uniform acceleration.
- Use the appropriate equations based on which variables are known and which are unknown.
- Assign positive or negative values to acceleration based on direction (speeding up or slowing down).
Example Problems
- Constant Velocity Example:
- Car travels 4 hours at 60 mph north: Displacement = (60 \times 4 = 240) miles.
- Average Speed (Round Trip) Example:
- Go 60 miles at 40 mph, return at 60 mph. Total time = (1.5 + 1 = 2.5) hours; average speed = (120/2.5 = 48) mph (not the simple average).
- Uniform Acceleration Example:
- Car starts from rest, (a = 10, m/s^2):
- Velocity after 6 s: (0 + 10 \times 6 = 60, m/s).
- Distance in each second increases:
- 0–1 s: 5 m, 1–2 s: 15 m, 2–3 s: 25 m.
Key Terms & Definitions
- Displacement ((\Delta x)) — Change in position; vector quantity.
- Velocity ((v)) — Rate of change of displacement; can have direction (vector).
- Acceleration ((a)) — Rate of change of velocity; vector.
- Uniform acceleration — Acceleration that remains constant over time.
- Average velocity — ((v_0 + v)/2) during constant acceleration.
- From rest — Initial velocity ((v_0)) is zero.
Action Items / Next Steps
- Practice using the four main kinematics equations with different variable combinations.
- Review concepts of displacement, velocity, and acceleration.
- Check study guides for additional problems and practice opportunities.