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Position-time, velocity-time, and acceleration-time graphs
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Graphs are powerful tools for visualizing and analyzing motion. The three main types are:
A position-time graph is a straight line passing through m and m. What is the velocity?
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The slope of a position-time graph gives the velocity.
Key features:
The slope of a velocity-time graph gives the acceleration.
Key features:
The area under a velocity-time graph gives the displacement.
For constant velocity (rectangle):
For constant acceleration (trapezoid or triangle):
Sign matters:
The area under an acceleration-time graph gives the change in velocity.
For constant acceleration:
| If you have... | To get... | Operation |
|---|---|---|
| Position-time | Velocity-time | Take slope (derivative) |
| Velocity-time | Acceleration-time | Take slope (derivative) |
| Velocity-time | Position-time | Find area (integral) |
| Acceleration-time | Velocity-time | Find area (integral) |
Speeding up: velocity and acceleration have the same sign
Slowing down: velocity and acceleration have opposite signs
Given information:
Find: Velocity
Solution: The velocity is the slope of the position-time graph.
Answer: The velocity is 2 m/s (constant).
Interpretation: The object moves in the positive direction at a steady 2 m/s. Since the slope is constant (straight line), there is no acceleration.
A velocity-time graph shows a horizontal line at m/s from to s. Find the displacement and acceleration.
Given information:
Find:
Part 1: Displacement Displacement = area under velocity-time graph
A velocity-time graph shows a straight line from to m/s. Find: (a) the acceleration, (b) the displacement during this time.
Given information:
Part (a): Find acceleration Acceleration = slope of velocity-time graph
Part (b): Find displacement Displacement = area under velocity-time graph
The graph forms a triangle with:
Alternative method for (b): Use kinematic equation:
Answers:
Check: Using : ✓
Since velocity is constant, the area is a rectangle:
Part 2: Acceleration Acceleration = slope of velocity-time graph
Since the line is horizontal:
Answers:
Key insight: Horizontal line on - graph means constant velocity (zero acceleration).