Describing Motion: Scalars, Vectors and Measurement

Every physical quantity is a number and a unit. The number alone is meaningless: $9.81$ is not an acceleration, $9.81{\text{m/s}}^2$ is. Canada uses SI units throughout, and this course leans on a handful of them often enough that you should know them without thinking.

Prefixes scale a unit by a power of ten — $\text{k}$ (kilo, $10^3$), $\text{c}$ (centi, $10^{-2}$), $\text{m}$ (milli, $10^{-3}$). Convert to base SI units before you substitute into a formula, not after. A speed left in km/h and dropped into an equation expecting m/s is the single most common source of an answer that is wrong by a factor of $3.6$.

A measurement carries its own precision. A stopwatch reading $4.2\text{s}$ is claiming something weaker than one reading $4.200\text{s}$. Significant figures are how you stop a calculator from inventing precision your instruments never had.

1. All non-zero digits count. $47.3$ has three.
2. Zeros between non-zero digits count. $5.008$ has four.
3. Leading zeros never count — they only place the decimal. $0.0042$ has two.
4. Trailing zeros count only if there is a decimal point. $4.50$ has three; $450$ is ambiguous and is normally read as two. Write $4.50\times 10^2$ if you mean three.

A scalar is fully described by a size: a mass of $70\text{kg}$, a time of $12\text{s}$. A vector needs a size and a direction: a velocity of $12\text{m/s}[\text{E}]$. Square brackets are the standard way to write that direction, and a vector answer without one is an incomplete answer.

Distance is how far you actually travelled — the length of the path, and it can only grow. Displacement is the straight-line change in position from start to finish, with a direction. Run one full lap of a $400\text{m}$ track and your distance is $400\text{m}$ while your displacement is zero, because you finished exactly where you started.

The same split runs through the rate quantities. Average speed is total distance over total time, a scalar. Average velocity is displacement over time, a vector pointing wherever the displacement pointed. The lap runner has a healthy average speed and an average velocity of exactly zero.

1. Initiating and planning — state a testable question and identify the independent variable (the one you change), the dependent variable (the one you measure) and the controlled variables (everything deliberately held constant). Predict an outcome, and say why.
2. Performing and recording — take repeated measurements, record every raw reading with its unit, and never quietly discard a reading you dislike.
3. Analysing and interpreting — graph it. Independent variable on the horizontal axis, dependent on the vertical. A straight-line graph beats a table, because its slope is usually the quantity you were actually after.
4. Communicating — report the result with correct significant figures, units and direction, and compare it with the accepted value: $\%\text{error}=\frac{|\text{measured}-\text{accepted}|}{\text{accepted}}\times 100\%\text{.}$