Fundamentals of mechanics: describing motion

Physics describes motion with two kinds of quantity. A scalar has magnitude only: distance, speed, time, mass and energy are scalars. A vector has magnitude and direction: displacement, velocity and acceleration are vectors. A vector is drawn as an arrow whose length is proportional to its magnitude and whose orientation shows its direction, and in one dimension we often use a sign convention such as east positive and west negative.

Distance is the total length of the path travelled, so it is never negative and never decreases. Displacement is the straight-line change in position from start to finish, together with its direction. If you walk around a block and end where you began, your distance is the perimeter but your displacement is zero. Displacement can be smaller than distance, and the two are equal in size only for straight-line motion that never reverses.

Speed is distance divided by time, and velocity is displacement divided by time. Average speed = total distance ÷ total time, while average velocity = displacement ÷ time taken, and it points in the direction of the displacement. Because they use different numerators, a person can have a high average speed yet an average velocity of zero. The SI unit for both is metres per second (m s⁻¹).

• Scalars add as ordinary numbers; vectors must be added with direction taken into account.
• Always state a direction (for example 'north', 'up the slope' or a positive sign) when giving a vector answer.
• Use a labelled arrow or a sign convention and keep it consistent through a whole problem.

Displacement over two perpendicular legs is found using Pythagoras' theorem for the magnitude and trigonometry for the direction. The angle is usually quoted from a reference direction, such as 'north of east', so that another person could draw your vector exactly.