Overview
This lecture covers solving kinematics problems involving displacement, distance, and turning points using constant acceleration equations, discusses a classic "chasing" problem with two moving objects, and introduces vertical (1D) motion under gravity.
Displacement and Turning Points (Homework 1.9 Parts C & D)
- An object moves left with a negative initial velocity and later right with a positive final velocity, indicating a turning point where velocity is zero.
- Displacement (Δx) is calculated as x_final − x_initial, regardless of the path taken or turning points.
- The kinematic equation used: v_f² = v_i² + 2aΔx.
- For distance, split the motion at the turning point: calculate Δx₁ (to turning point) and Δx₂ (from turning point onward).
- Total distance = |Δx₁| + |Δx₂|; distance ignores direction, only magnitude.
- On a velocity-time (v-t) graph, area under the line (considering sign) represents displacement; sum of absolute areas gives distance.
- Displacement equals the sum of Δx₁ and Δx₂ (with signs), while distance is the sum of their magnitudes.
Chasing Problem (Homework Question 10)
- Two people: Person A at rest (starts with 0 velocity, constant acceleration after a delay), Person B moves at constant speed.
- Both share a synchronized clock; Person A starts chasing after Person B passes by.
- At the catch-up point, Person A and B have the same final position.
- Write equations for positions:
- Person A: x_A = (1/2)·a·t² (starts from rest)
- Person B: x_B = v_B·t_delay + v_B·t
- Set x_A = x_B to form a quadratic equation in t (the catch-up time).
- Solve for t using the quadratic formula; discard non-physical (negative) roots.
- Distance traveled by A is simply x_A using calculated t.
Introduction to Vertical Motion (Section 8)
- Now considering motion under gravity: an object thrown vertically with initial velocity.
- Only force acting is gravity (constant acceleration, a_y = −9.8 m/s²).
- At the highest point, velocity is zero (turning point); after, object moves down.
- The four kinematic equations still apply, with appropriate axis and sign convention (up is positive).
- Will rewrite kinematic equations for vertical (y) motion in next lecture.
Key Terms & Definitions
- Displacement (Δx) — The straight-line change in position; x_final − x_initial.
- Distance — Total path length traveled, regardless of direction (sum of absolute displacements).
- Turning Point — Moment when velocity changes sign (often where an object stops briefly).
- Constant Acceleration — Acceleration that does not change over time.
- Kinematic Equations — Four standard equations relating position, velocity, acceleration, and time for constant acceleration.
- v-t Graph — Velocity versus time graph; area under the curve equals displacement.
Action Items / Next Steps
- Complete homework problems 1–11 (up to the end of Section 7).
- Begin reading Section 8 ("3D Falling Motion") in the textbook.
- Watch the video derivation of the four kinematic equations if you missed the previous lecture.