SPI Physics · Kinematics and Motion

Distance, Displacement and Speed

Motion is always measured against something; distance is the length of the path, displacement is the straight line from start to finish, and average speed is total distance over total time.

All Kinematics and Motion notes

Why this matters

Three questions in the bank, two EASY and one MODERATE: a passenger in a train, the displacement on a circular path, and the speed needed to reach a target average. The circle and the average are where marks go, because the tempting answers (the arc length, the mean of two speeds) are printed as options.

Concept 1 of 3: Rest and motion depend on what you compare with

A book on a seat in a bus does not move compared with the seat, but it does move compared with the road. So 'at rest' or 'in motion' only means something once you say compared with what.

Definition

A body is at rest if its position does not change compared with its surroundings, and in motion if it does.
The thing you compare with is the reference point (or frame of reference).
So rest and motion are relative: the same body can be at rest compared with one thing and in motion compared with another, at the same time.

Body	At rest compared with	In motion compared with
A passenger sitting in a moving bus	The bus and the other passengers	The road and the trees outside
A tree by the road	The ground	A passenger in a passing bus
A pilot flying a plane	The plane	The ground

Answer 'rest or motion?' with both, once you name the two reference points.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Kinematics and Motion · Distance, Displacement and Speed

A passenger is sitting in a moving train the person is

Picking only one of rest and motion

A body can be at rest and in motion at the same time, compared with two different things. When the options include 'both', that is the answer for a passenger in a moving vehicle.

Concept 2 of 3: Distance and displacement

Distance is what the odometer counts: every metre of the path. Displacement only cares where you started and where you stopped, and measures the straight line between them with a direction.

Definition

Distance is the length of the actual path. It has size only, and it can never be zero once the body has moved.
Displacement is the straight-line length from the start to the end, in a given direction.
Displacement is never more than distance. They are equal only when the body moves in a straight line without turning back.
Back at the start, displacement is zero, however long the path.
On a circle of radius r, one full round is a distance of $2\pi r$ and a displacement of 0.

Distance and displacement

\text{displacement} \le \text{distance} \qquad \text{circumference of a circle} = 2\pi r

Worked example

A boy walks 3 km east and then 4 km north. Find the distance he walks and his displacement.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Kinematics and Motion · Distance, Displacement and Speed

A particle is moving in a circular path of radius r. The displacement after half a circle would be

Giving the arc length as the displacement

On a circular path the arc is the distance. The displacement is the straight chord from start to end: across the full width of the circle, it is the diameter, 2r.

Thinking a full round is far away

After one complete round the body is back where it began, so the displacement is zero, even though the distance is

2\pi r

Concept 3 of 3: Average speed over two or more parts of a journey

Average speed is not the average of the speeds. It is one total divided by another: how far you went altogether, over how long it took altogether.

Definition

Speed $=$ distance $\div$ time. Its SI unit is m/s.
Average speed $=$ total distance $\div$ total time, adding up every part of the journey.
Change minutes into hours before using km/h: 20 minutes is $\frac{1}{3}$ h.
To change km/h into m/s, multiply by $\frac{5}{18}$ ; to go back, multiply by $\frac{18}{5}$ .
To find the speed needed for a target average: total distance needed $=$ target average $\times$ total time; subtract what is already covered; divide by the time left.

Average speed

\text{average speed} = \frac{\text{total distance}}{\text{total time}} \qquad 1\ \text{km/h} = \frac{5}{18}\ \text{m/s}

Worked example

A car goes 60 km to a town at 30 km/h and comes back the same 60 km at 60 km/h. Find its average speed for the whole trip.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Kinematics and Motion · Distance, Displacement and Speed

A boy runs for 10 minutes at a uniform speed of 9

\text{km h}^{-1}

. At what speed should he run for the next 20 minutes so that the average speed becomes 12

\text{km h}^{-1}

Averaging the two speeds

When the times are different, the mean of the two speeds is wrong. Find the total distance and the total time first; only equal TIMES make the simple mean right.

Using minutes with km/h

15 minutes at 8 km/h is

8 \times \frac{1}{4} = 2

km, not 120 km. Turn minutes into hours (divide by 60) before multiplying.

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 (2)

Distance and displacement
Distance and displacement
$\text{displacement} \le \text{distance} \qquad \text{circumference of a circle} = 2\pi r$
Average speed over two or more parts of a journey
Average speed
$\text{average speed} = \frac{\text{total distance}}{\text{total time}} \qquad 1\ \text{km/h} = \frac{5}{18}\ \text{m/s}$

Reference tables (1)

Rest and motion depend on what you compare with3 rows

Body	At rest compared with	In motion compared with
A passenger sitting in a moving bus	The bus and the other passengers	The road and the trees outside
A tree by the road	The ground	A passenger in a passing bus
A pilot flying a plane	The plane	The ground

Answer 'rest or motion?' with both, once you name the two reference points.

Watch out for (5)

Picking only one of rest and motion
→ Rest and motion depend on what you compare with
Giving the arc length as the displacement
→ Distance and displacement
Thinking a full round is far away
→ Distance and displacement
Averaging the two speeds
→ Average speed over two or more parts of a journey
Using minutes with km/h
→ Average speed over two or more parts of a journey

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