IMAT Physics · Kinematics

Displacement, Velocity, Relative Motion and Acceleration

Velocity is how fast position changes, with a direction; acceleration is how fast velocity changes; and velocities measured by moving observers add like vectors.

All Kinematics notes

Why this matters

Six of the eight past questions are on this page, all four ministry ones among them: a definition of velocity in 2026, the average velocity of a two-leg trip in 2023, one car overtaking another in 2025, and when Galilean transformations hold in 2026. The older papers asked for a displacement after four legs and a boat crossing a flowing river.

Concept 1 of 3: Distance and displacement, speed and velocity

Distance counts every step you take; displacement only cares where you end up compared with where you started, and in which direction. Speed and velocity follow the same split. A runner who finishes a lap where she began has covered a long distance at a good speed, yet her displacement and her average velocity are both zero.

Definition

Distance (scalar): the length of the path. Displacement (vector): the straight line from start to finish, with its direction.
Average speed = total distance ÷ total time. Average velocity = displacement ÷ total time.
Velocity is the rate of change of position: it says how quickly, and in which direction, a body changes its position.
In uniform motion (constant velocity in a straight line), $s = vt$ .
Unit change: $1\ \text{m/s} = 3.6\ \text{km/h}$ . Divide km/h by 3.6 to get m/s.

Average speed and average velocity

\text{average speed} = \frac{\text{distance}}{t} \qquad \bar{v} = \frac{\Delta s}{t}

$\bar{v}$ average velocity, a vector
$\Delta s$ displacement, from start to finish, in m
$t$ total time taken, in s

Worked example

A dog runs 30 m east to fetch a ball and then 40 m west, taking 10 s in all. Find its average speed and its average velocity.

Practice this conceptself-check · 4 quick reps

Average speed is not the average of the speeds

Average speed is total distance over total time. Going 60 km at 60 km/h and 60 km back at 30 km/h gives 40 km/h, not 45 km/h, because more time is spent at the lower speed.

Average velocity uses the straight-line displacement

For a trip with legs at right angles, the displacement comes from Pythagoras, not from adding the legs. Dividing the total path by the time gives the average speed, and IMAT lists it as a wrong option.

Concept 2 of 3: Relative velocity and Galilean transformations

Velocity is always measured by someone. A passenger walking down a moving train has one velocity relative to the train and another relative to the ground. As long as one observer moves at a constant velocity in a straight line relative to the other, you get from one measurement to the other by simply adding or subtracting that velocity.

Definition

The velocity of A relative to B is $v_{AB} = v_A - v_B$ (vectors).
Same direction: the relative speed is the difference of the speeds. Opposite directions: it is the sum.
A boat or plane moving through water or air that itself moves: its velocity over the ground is the vector sum of its velocity through the medium and the medium's velocity.
Galilean transformations ( $x' = x - ut$ , $v' = v - u$ ) link two observers in uniform rectilinear motion relative to each other (constant velocity, straight line). They do not hold if one observer accelerates or rotates.
Both observers then measure the same acceleration for any object. (At speeds near that of light, relativity replaces these rules.)

Relative velocity

v_{AB} = v_A - v_B \qquad t_{\text{catch}} = \frac{\text{gap}}{v_A - v_B}

$v_A, v_B$ velocities of A and B measured by the same observer
$t_{\text{catch}}$ time for A to close a gap on B ahead of it, moving the same way

Worked example

A bus leaves a stop at a steady 15 m/s. A car passes the same stop 20 s later at a steady 25 m/s on the same road. How long after passing the stop does the car catch the bus?

Practice this conceptself-check · 4 quick reps

Galilean transformations need uniform straight-line relative motion

They hold only when one observer moves at a constant velocity in a straight line relative to the other. Accelerating, braking or rotating observers are excluded, even when the rotation is at a constant rate.

Check the units before comparing a gap with a speed

When speeds are in km/h and a gap is in metres, convert first. A gap of 400 m closed at 36 km/h (10 m/s) takes 40 s; dividing 400 by 36 gives a meaningless 11.

Concept 3 of 3: Acceleration: the rate of change of velocity

Acceleration tells you how quickly the velocity is changing. Since velocity has a direction, a body accelerates when it speeds up, slows down, or simply turns. A car cruising at a steady 100 km/h on a straight road has zero acceleration, however fast it goes.

Definition

Acceleration is the change of velocity per unit time, a vector measured in $\text{m/s}^2$ .
Slowing down is an acceleration opposite to the velocity (a negative value if forwards is positive), often called deceleration.
Uniform motion means constant velocity, so zero acceleration. Uniformly accelerated motion means the acceleration is constant.
A body can have zero velocity and still be accelerating (a ball at the top of its flight).

Average acceleration

a = \frac{\Delta v}{\Delta t} = \frac{v - u}{t}

$u$ initial velocity, in m/s
$v$ final velocity, in m/s
$t$ time taken, in s

Worked example

A car speeds up from 8.0 m/s to 26 m/s in 6.0 s on a straight road. What is its acceleration?

Practice this conceptself-check · 3 quick reps

Zero velocity does not mean zero acceleration

At the top of its flight a ball thrown straight up is momentarily still, but gravity is still changing its velocity at

9.8\ \text{m/s}^2

downwards. Acceleration depends on how velocity is changing, not on its value.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Distance and displacement, speed and velocity
Average speed and average velocity
$\text{average speed} = \frac{\text{distance}}{t} \qquad \bar{v} = \frac{\Delta s}{t}$
Relative velocity and Galilean transformations
Relative velocity
$v_{AB} = v_A - v_B \qquad t_{\text{catch}} = \frac{\text{gap}}{v_A - v_B}$
Acceleration: the rate of change of velocity
Average acceleration
$a = \frac{\Delta v}{\Delta t} = \frac{v - u}{t}$

Watch out for (5)

Average speed is not the average of the speeds
→ Distance and displacement, speed and velocity
Average velocity uses the straight-line displacement
→ Distance and displacement, speed and velocity
Galilean transformations need uniform straight-line relative motion
→ Relative velocity and Galilean transformations
Check the units before comparing a gap with a speed
→ Relative velocity and Galilean transformations
Zero velocity does not mean zero acceleration
→ Acceleration: the rate of change of velocity

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank