Displacement and Distance

Definition: Distance

Distance is the total length of the path taken in going from the initial position, \(\vec{x}_{i}\), to the final position, \(\vec{x}_{f}\). Distance is a scalar.

Quantity: Distance (\(D\)) Unit name: metre Unit symbol: \(\text{m}\)

Tip:

The symbol Δ is read out as delta. Δ is a letter of the Greek alphabet and is used in Mathematics and Science to indicate a change in a certain quantity, or a final value minus an initial value. For example, \(\Delta x\) means change in \(x\) while \(\Delta t\) means change in \(t\).

In the simple map below you can see the path that winds because of a number of hills from a school to a nearby shop. The path is shown by a dashed line. The initial point, \(\vec{x}_{i}\), is the school and the final point, \(\vec{x}_{f}\), is the shop.

Distance is the length of dashed line. It is how far you have to walk along the path from the school to the shop.

Definition: Displacement

Displacement is the change in an object's position. It is a vector that points from the initial position (\(\vec{x}_{i}\)) to the final position (\(\vec{x}_{f}\)).

Quantity: Displacement (\(\Delta \vec{x}\)) Unit name: metre Unit symbol: \(\text{m}\)

The displacement of an object is defined as its change in position (final position minus initial position). Displacement has a magnitude and direction and is therefore a vector. For example, if the initial position of a car is \(\vec{x}_{i}\) and it moves to a final position of \(\vec{x}_{f}\), then the displacement is:

\[\Delta \vec{x} = \vec{x}_{f} - \vec{x}_{i}\]

To help visualise what the displacement vector looks like think back to the tail-to-head method. The displacement is the vector you add to the initial position vector to get a vector to the final position.

However, subtracting an initial quantity from a final quantity happens often in Physics, so we use the shortcut Δ to mean final - initial. Therefore, displacement can be written:

\[\Delta \vec{x} = \vec{x}_{f} - \vec{x}_{i}\]

The following diagram illustrates the concept of displacement:

For example, if you roll a ball \(\text{5}\) \(\text{m}\) along a floor, in a straight line, then its displacement is \(\text{5}\) \(\text{m}\), taking the direction of motion as positive, and the initial position as \(\text{0}\) \(\text{m}\).

Tip:

The words initial and final will be used very often in Physics. Initial refers to the situation in the beginning of the description/problem and final to the situation at the end. It will often happen that the final value is smaller than the initial value, such that the difference is negative. This is ok!

Displacement does not depend on the path travelled, but only on the initial and final positions. We use the word distance to describe how far an object travels along a particular path.

If we go back to the simple map repeated below you can see the path as before shown by a dashed line.

Tip:

We will use \(D\) in this book, but you may see \(d\) used in other books.

Tip:

We use the expression 'as the crow flies' to mean a straight line between two points because birds can fly directly over many obstacles.

Distance is the length of dashed line. The displacement is different. Displacement is the straight-line distance from the starting point to the endpoint – from the school to the shop in the figure as shown by the solid arrow.

Displacement and Distance