Describing motion: distance, displacement, speed, velocity and acceleration

Scalars and vectors

• A scalar has size only.
  • Examples: distance, speed, time, mass, energy.
• A vector has size and direction.
  • Examples: displacement, velocity, acceleration, force, momentum.
• Direction is stated with a word (north, upward, left) or a sign.
  • You choose which direction is positive. Everything in the opposite direction is then negative.
  • Once chosen, keep the same convention for the whole question.

Distance and displacement

• Distance ($d$) is how far something has travelled along its path. It is a scalar and never decreases.

• Displacement is how far it ends up from where it started, in a stated direction. It is a vector.

• The two are only equal when the motion is in a straight line without turning back.

• A runner completing one lap of a $400$ m track has:

  • travelled a distance of $400$ m,
  • a displacement of zero, because the finish point is the start point.

Speed and velocity

• Speed is the rate of covering distance — a scalar.
• Velocity ($v$) is the rate of change of displacement — a vector, so it has a direction.

$$v = \frac{d}{t}$$

• $v$ — velocity (m s⁻¹)

• $d$ — distance or displacement (m)

• $t$ — time taken (s)

• Average velocity uses the whole journey: total displacement ÷ total time.

• Instantaneous velocity is the velocity at one moment — what a speedometer shows.

  • These differ whenever the motion changes. A car that speeds up and slows down has an average velocity that it may never actually travel at.

Acceleration

• Acceleration ($a$) is the rate of change of velocity. It is a vector.

$$a = \frac{\Delta v}{t} = \frac{v_f - v_i}{t}$$

• $a$ — acceleration (m s⁻²)

• $\Delta v$ — change in velocity (m s⁻¹), always final minus initial

• $v_i$ — initial velocity (m s⁻¹)

• $v_f$ — final velocity (m s⁻¹)

• $t$ — time over which the change happens (s)

• Accelerating means the velocity is changing. That covers three different things:

  • speeding up — acceleration in the same direction as the velocity,
  • slowing down — acceleration opposite to the velocity (often called deceleration),
  • changing direction at constant speed — as in circular motion.

• A negative acceleration does not always mean slowing down. It means the acceleration points in the negative direction. An object already travelling in the negative direction with a negative acceleration is speeding up.

Choosing and using a sign convention

• Write your positive direction down at the top of the answer, e.g. "taking up as positive".
• Substitute the signs, then let the algebra decide.
  • A ball thrown up at $12$ m s⁻¹ with up positive has $v_i = +12$ m s⁻¹ and $a = -9.8$ m s⁻².
• Read the sign of the answer — it tells you the direction, and it is part of the answer.