Distance and displacement

Two ways to count a walk

You walk to a shop $300\ \text{m}$ away and come back. You walked $600\ \text{m}$. But where are you now? Right where you started. Physics keeps both facts, with two names.

• Distance is the length of the path you took. It counts every step, never goes down, and has no direction.
• Displacement is the straight line from the start to the end, with its direction. The route in between does not matter.

For the trip to the shop and back, the distance is $600\ \text{m}$ and the displacement is $0\ \text{m}$.

A walk with a turn

Meena walks $3\ \text{m}$ east and then $4\ \text{m}$ north.

Her distance is $3\ \text{m} + 4\ \text{m} = 7\ \text{m}$. Her displacement is the long side of a right-angled triangle: $\sqrt{(3\ \text{m})^2 + (4\ \text{m})^2} = 5\ \text{m}$, a little east of north.

You should find that the displacement is shorter than the distance whenever the walk turns. They are equal only when you walk in one straight line without turning back.

Watch out. On a straight road, use signs. Going $10\ \text{km}$ north and then $4\ \text{km}$ south gives a displacement of $10\ \text{km} - 4\ \text{km} = 6\ \text{km}$ north, not $14\ \text{km}$.

A worked example

Arjun walks $6\ \text{m}$ east and then $8\ \text{m}$ north. Find his distance and his displacement.

1. Distance. $6\ \text{m} + 8\ \text{m} = 14\ \text{m}$.
2. Displacement. $\sqrt{(6\ \text{m})^2 + (8\ \text{m})^2} = \sqrt{100\ \text{m}^2} = 10\ \text{m}$.
3. So he walked $14\ \text{m}$ but is only $10\ \text{m}$ from his start.

Practise with new numbers

Try it in Python

The program adds up a walk step by step.

Remember these

• Distance: the length of the path. Displacement: start to end, with a direction.
• The displacement is never longer than the distance.
• For a walk with one right-angle turn, displacement $= \sqrt{a^2 + b^2}$.

Check yourself

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

1. A runner does one full lap of a $400\ \text{m}$ track. What are her distance and displacement?

2. Can the displacement be larger than the distance?

3. A car goes $9\ \text{km}$ east and $2\ \text{km}$ back west. Find its distance and displacement.

4. A taxi fare depends on which one, distance or displacement?