Position, path length and displacement

Where is the bus?

A friend calls and asks, "Where are you?" You say, "On the main road." That does not help much. She needs to know where on the road, and measured from what. "$300\ \text{m}$ east of the school gate" does help.

So to give the place of anything, we need two things: a starting point and a direction. In this chapter every body moves along one straight line, so that is all we need.

An axis on the road

Draw a straight line along the road and choose:

• a point on it, the origin $O$, where $x = 0$;
• a positive direction, usually to the right, east, or up.

The position $x$ of a body is how far it is from $O$, with a sign. A body $30\ \text{m}$ to the right of $O$ is at $x = +30\ \text{m}$. One $30\ \text{m}$ to the left is at $x = -30\ \text{m}$.

You may put the origin anywhere and point the axis either way. The physics does not change; only the numbers do. A clever choice makes a problem easy, so we often put the origin where the motion starts.

The origin, the axis and a clock together make a frame of reference. A body moves if its position in this frame changes with time.

Can a train be a point? If a body is small compared with how far it goes, we treat it as a single point. A train going $500\ \text{km}$ can be a point. The same train in a station, when we ask which coach is at the sign, cannot.

Two ways to say "how far"

You walk from home to a shop $80\ \text{m}$ to the right. Then you walk $50\ \text{m}$ back, to a bank.

• The path length (or distance) is the length of the whole walk: $80\ \text{m} + 50\ \text{m} = 130\ \text{m}$. It is a scalar: a size with no direction. It is never negative, and it never goes down as time goes on.
• The displacement is the change of position:

Here $\Delta x = (+30\ \text{m}) - 0\ \text{m} = +30\ \text{m}$. The plus sign says you ended to the right of where you began. Displacement is a vector; along a line, its sign is its direction.

Play with it

Move the three positions. Watch both numbers.

You should find:

• The displacement depends only on the start and the end. The path length depends on the whole route.
• The displacement can be positive, negative or zero. The path length never is negative.
• The size of the displacement is never more than the path length. They are equal only when the body never turns back.

A worked example

A shunting engine starts at the $2\ \text{km}$ post of a straight track, runs forward to the $9\ \text{km}$ post, and then reverses to the $5\ \text{km}$ post. Take the station as the origin.

1. First leg. From $2\ \text{km}$ to $9\ \text{km}$: $7\ \text{km}$ forward.
2. Second leg. From $9\ \text{km}$ to $5\ \text{km}$: $4\ \text{km}$ back.
3. Path length. $7\ \text{km} + 4\ \text{km} = 11\ \text{km}$.
4. Displacement. $5\ \text{km} - 2\ \text{km} = +3\ \text{km}$, forward.

Watch out. The displacement is not the final position, $+5\ \text{km}$. That would be true only if the engine had started at the station.

Think first. Would the answers change if we measured from the $2\ \text{km}$ post instead of the station? The positions would become $0\ \text{km}$, $7\ \text{km}$ and $3\ \text{km}$. The path length is still $11\ \text{km}$ and the displacement still $+3\ \text{km}$: differences do not depend on the origin.

Practise with new numbers

Try it in Python

The program adds up the legs of the engine's trip, and finds its displacement from the first and last positions.

Remember these

• Position needs an origin and a positive direction: $x$ has a sign.
• Path length: every metre travelled, never negative.
• Displacement: $\Delta x = x_{\text{final}} - x_{\text{initial}}$, with a sign, never larger in size than the path length.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. You walk $120\ \text{m}$ east along a straight road, then $45\ \text{m}$ west. Find your path length and your displacement.

2. Give an example in which the displacement is zero but the path length is not.

3. Your friend measures positions from the school gate. You measure them from your home, $200\ \text{m}$ east of the gate. Do you agree on your positions? On your displacements?

4. A body has a path length of $50\ \text{m}$ and a displacement of $-50\ \text{m}$. What can you say about its motion?