Define displacement, velocity and acceleration for motion in a straight line and interpret their signs and the graphs that connect them

What this dot point is asking

SEAB wants you to use the precise meanings of displacement, velocity and acceleration for a particle moving along a straight line, to read the information their signs carry (direction, speeding up or slowing down), and to interpret displacement-time and velocity-time graphs. This vocabulary underpins all of the kinematics calculus that follows.

The answer

The three quantities

For motion along a line, measured from a fixed origin $O$:

• Displacement $s$ is the position relative to $O$, with sign showing direction. It is not the same as distance travelled.
• Velocity $v$ is the rate of change of displacement, with sign showing the direction of motion. Its magnitude is the speed.
• Acceleration $a$ is the rate of change of velocity.

What the signs mean

A positive displacement is one side of $O$, negative the other. A positive velocity means moving in the positive direction; negative means the other way. The particle is instantaneously at rest when $v = 0$, which is where it can change direction.

Speeding up or slowing down

The particle speeds up when velocity and acceleration have the same sign, and slows down when they have opposite signs. So a negative velocity with a positive acceleration means the particle is slowing as it moves in the negative direction.

Reading the graphs

On a displacement-time graph, the gradient is the velocity. On a velocity-time graph, the gradient is the acceleration and the area under the graph is the displacement. A horizontal displacement-time graph means the particle is at rest; a velocity-time graph crossing the axis marks a change of direction.

Examples in context

Example 1. A ball thrown upward. A ball thrown up has positive velocity that decreases to zero at the top (where it is momentarily at rest), then negative velocity coming down, while the acceleration stays negative throughout, illustrating opposite signs on the way up and matching signs on the way down.

Example 2. A train between stations. A train accelerates from rest, travels at constant velocity, then decelerates to stop; its velocity-time graph is a trapezium whose area gives the distance between stations, a direct reading of displacement from the graph.

Try this

Q1. A particle has $s = 5t - t^2$. Find its velocity at $t = 2$. [2 marks]

• Cue. $v = 5 - 2t = 5 - 4 = 1\ \text{m s}^{-1}$.

Q2. State what $v = 0$ tells you about the motion. [1 mark]

• Cue. The particle is instantaneously at rest, a possible turning point.

Q3. A particle has $v = t^2 - 4$. Find the acceleration when $t = 3$. [2 marks]

• Cue. $a = \dfrac{dv}{dt} = 2t = 6\ \text{m s}^{-2}$.