Units and Quantities

Units and Quantities

Fundamental Units

  • Fundamental units are the basic building blocks of quantities and comprise length, mass, and time with their respective units: metre (m), kilogram (kg), and second (s).
  • All other units can be expressed as combinations of these fundamental units.

Derived Units

  • Derived units are developed from the fundamental units. For example, speed is calculated as the change in position (metres, m) per change in time (seconds, s) and so its unit is metres per second (m/s).
  • Another example is acceleration, which is the change in speed per unit time. Its unit is metres per second squared (m/s²).

Scalars and Vectors

  • Scalar quantities have magnitude only. Examples include distance, speed, mass, temperature and energy. They can be added, subtracted, multiplied and divided in the normal way.
  • Vector quantities, on the other hand, have both magnitude and direction. Examples include displacement, velocity, acceleration, and force. Vector addition and subtraction involve both the magnitudes and the directions of the vectors.

Methods of Describing Quantities

  • Equations are often used to describe quantities; for instance, distance is equal to speed times time (d = s x t).
  • Diagrams can also help visualise quantities; force diagrams, for example, can efficiently represent forces acting on an object.
  • Graphs are another tool for depicting quantities where the relationship between two quantities is shown; for example, a distance-time graph can describe a journey, with the slope representing speed.

Changing Units

  • It is important to be consistent with units throughout calculations.
  • Conversion between units requires an understanding of how units relate to each other, e.g., 1 km equals 1000 m.
  • Care should be taken with squared or cubed units; for instance, 1 km² is not equal to 1000 m² but equals 1,000,000 m².

Estimates and Approximations

  • Estimating is often needed when exact values are not achievable or necessary. This is a useful skill, particularly for checking the reasonableness of solutions or when working with rough data.
  • Approximations can be made to simplify mathematical treatments; for example, treating an irregularly shaped object as a regular shape such as a sphere or a cylinder.

Important Concepts

  • Familiarity with the units of quantities often provides clues about the processes involved in solving a problem.
  • Units also serve as a check on the correctness of calculations. If the units of the answer do not match what is expected, this can indicate an error has been made in the calculations.
  • Practice using different units and converting between them. This will improve understanding of how units integrate with quantities and equations.