Overview
This lecture explains how to interpret motion graphs in physics, focusing on the meaning of slope and area for position, velocity, and acceleration vs. time graphs.
Slope and Area Concepts
- Slope equals change in y divided by change in x (division).
- Area under a graph equals y multiplied by x (multiplication).
- Slope formula: (y₂ - y₁) / (x₂ - x₁); Area formula: length × width (rectangle).
Position-Time Graph (x vs. t)
- Slope of position-time graph = velocity (rate of change of position).
- Slope at a point (tangent) = instantaneous velocity; between two points (secant) = average velocity.
- Area under position-time graph does not have a useful physical meaning.
Velocity-Time Graph (v vs. t)
- Slope of velocity-time graph = acceleration (rate of change of velocity).
- Area under velocity-time graph = displacement (change in position).
Acceleration-Time Graph (a vs. t)
- Slope of acceleration-time graph = "jerk/jolt" (rate of change of acceleration, rarely used in basic physics).
- Area under acceleration-time graph = change in velocity (v_final - v_initial).
Interpreting Graphs: Tangent vs. Secant
- Slope of a tangent line on position-time graph gives instantaneous velocity.
- Slope of secant line gives average velocity.
- Approximate instantaneous slope by using secant lines close to the desired point.
Position vs. Distance-Time Graphs
- Slope of position-time graph = velocity (vector, can be negative or positive).
- Slope of distance-time graph = speed (scalar, always positive).
- Velocity = displacement/time; Speed = distance/time.
Signs and Motion
- Positive velocity = moving right; negative = moving left.
- Zero velocity: object may be at rest or changing direction.
- Positive acceleration: increasing velocity; negative acceleration: decreasing velocity.
- Speed = absolute value of velocity (always positive).
Speeding Up vs. Slowing Down
- Object speeds up when velocity and acceleration have the same sign.
- Object slows down when velocity and acceleration have opposite signs.
Linear and Parabolic Position-Time Graphs
- Linear graphs: slope is constant, velocity is constant, acceleration is zero.
- Upwards: positive velocity; flat: zero velocity; downwards: negative velocity.
- Parabolic graphs: velocity is changing, so acceleration is nonzero.
- Concave down: negative acceleration; concave up: positive acceleration.
- Analyze slope changes to determine acceleration direction.
- Signs of velocity and acceleration determine if speeding up or slowing down.
Key Terms & Definitions
- Slope — Change in y divided by change in x; gives rate of change.
- Area under graph — Product of y and x values; gives accumulated quantity.
- Velocity — Rate of change of position; vector quantity.
- Speed — Magnitude of velocity; scalar quantity.
- Acceleration — Rate of change of velocity.
- Displacement — Change in position.
- Instantaneous velocity — Slope of tangent line at a point.
- Average velocity — Slope between two points (secant line).
- Jerk/Jolt — Rate of change of acceleration.
Action Items / Next Steps
- Memorize which physical quantity is represented by slope and area for each type of motion graph.
- Practice identifying speeding up/slowing down by comparing acceleration and velocity signs.
- Review equations: x_final = x_initial + vt, v_final = v_initial + at.