Describing Motion in One Dimension: Quick Revision

A brief note on Describing Motion in One Dimension. Read it before a test, or whenever a motion question has you stuck. Try each question at the end before you open its answer.

The big ideas

• Position needs an origin and a positive direction, so $x$ has a sign.
• Path length adds up every metre travelled. It is never negative and never goes down.
• Displacement is the change of position, $\Delta x = x_{\text{final}} - x_{\text{initial}}$. It can be positive, negative or zero, and its size is never more than the path length.
• Average speed uses the path length; average velocity uses the displacement.
• Velocity at an instant is the slope of the tangent to the position-time graph, $v = \dfrac{dx}{dt}$. Its size is the speed.

The formulas at a glance

• Displacement: $\Delta x = x_2 - x_1$
• Average speed: $\dfrac{\text{path length}}{\Delta t}$
• Average velocity: $\dfrac{\Delta x}{\Delta t}$
• Velocity at an instant: $v = \dfrac{dx}{dt}$
• Two equal distances at $v_1$ and $v_2$: $\bar{v} = \dfrac{2v_1v_2}{v_1 + v_2}$

Position-time graphs in one look

• A flat line: at rest.
• A straight rising line: a steady positive velocity. A straight falling line: a steady negative velocity.
• A curve: the velocity changes. Steeper means faster.
• Two lines that cross: the two bodies are at the same place at the same time.

A worked example

A boy cycles from home $2.4\ \text{km}$ east to school in $12$ minutes, and then $0.9\ \text{km}$ back west to a shop in $6$ minutes.

1. Path length. $2.4\ \text{km} + 0.9\ \text{km} = 3.3\ \text{km}$. Displacement. $2.4\ \text{km} - 0.9\ \text{km} = 1.5\ \text{km}$ east.
2. Time. $18\ \text{min} = 0.3\ \text{h}$.
3. Average speed. $\dfrac{3.3\ \text{km}}{0.3\ \text{h}} = 11\ \text{km/h}$. Average velocity. $\dfrac{1.5\ \text{km}}{0.3\ \text{h}} = 5\ \text{km/h}$, east.

Practice with new numbers

A program to try

The program reads a table of positions and tells the story of the motion, stretch by stretch.

Mistakes to avoid

• The final position as the displacement. That works only if the body started at the origin.
• Averaging two speeds. Always divide the whole distance by the whole time.
• Reading the height of a position-time graph as the speed. The height is the position; the slope is the velocity.
• Average speed smaller than average velocity. It can never be: the path length is never less than the size of the displacement.

Check yourself

1. A lift goes from the ground floor up $24\ \text{m}$ and then down $9\ \text{m}$. Taking up as positive, what are its path length and its displacement?

2. A position-time graph is a straight line from $(0, 50\ \text{m})$ to $(10\ \text{s}, 20\ \text{m})$. What is the velocity?

3. For $x = 4t^2 + t$ (SI units), find the velocity at $t = 2\ \text{s}$.

4. Can a body's average velocity be zero while its average speed is not? Give an example.

5. Two lines on a position-time graph are parallel. What does that tell you about the two bodies?