Representing Motion
Introduced| English |
|---|
| slope/sləʊp/ |
| area/ˈeərɪə/ |
A whole journey in one picture
- A single graph can capture a whole trip -- where you were, how fast, speeding up or slowing.
- Learn to read the graph and you barely need equations.
- The trick is knowing what a slope and an area mean on each kind of graph.
- Master this and motion problems become almost visual.
Position-time graphs
- On a position-time graph, the slope 斜率 is the velocity.
- Steep means fast; flat means stopped; downward means moving backward.
- A curving line means the velocity is changing -- there is acceleration.
On a position-versus-time graph, the slope represents the...
Slope of position vs time is $dx/dt$ -- the velocity.
Velocity-time graphs
- On a velocity-time graph, the slope is the acceleration.
- The area 面积 under the line is the displacement.
- A flat line means constant velocity (zero acceleration).
Motion graphs: v and a
Change the acceleration and watch how the velocity-time and position-time graphs respond.
A velocity-time graph is a flat line at $5\ \tfrac{\text{m}}{\text{s}}$ for $6\ \text{s}$. What displacement does it show (in m)?
Displacement is the area under the graph, $5 \times 6 = 30\ \text{m}$.
On a velocity-time graph, the slope gives the ____.
Slope of velocity vs time is $dv/dt$ -- the acceleration.
Reading the shape
- Where a position graph is steepest, the object moves fastest.
- Where a velocity graph crosses zero, the object momentarily stops.
- The peaks and slopes tell the story without a single number.
On a velocity-time graph, the object is momentarily at rest where the line...
Velocity zero means momentarily at rest -- where the v-t line crosses the time axis.
A flat (horizontal) line on a position-time graph means the object is not moving.
Flat position means the position is not changing -- zero slope, zero velocity, so it is at rest.
Graphs mirror the calculus
- Slopes take you down the chain ($x \to v \to a$) -- that is differentiation.
- Areas take you up the chain ($a \to v \to x$) -- that is integration.
- The graph picture and the calculus are the same idea in two forms.
Select all correct pairings.
Slopes differentiate and areas integrate. The area under a position graph has no standard meaning.
A velocity-time graph is a flat line at $3\ \tfrac{\text{m}}{\text{s}}$ for $4\ \text{s}$.
- Slope $= 0$, so the acceleration is zero.
- Area $= 3 \times 4 = 12\ \text{m}$, so the displacement is $12\ \text{m}$.
Do not mix up the graphs. A flat line means "constant" on both -- but on a position graph it means not moving, while on a velocity graph it means moving at steady speed. Always check which quantity is on the vertical axis first.
On a position-time graph, the slope is velocity; on a velocity-time graph, the slope is acceleration and the area is displacement. Slopes differentiate ($x \to v \to a$); areas integrate ($a \to v \to x$). The graph and the calculus tell the same story.