Displacement: The direction and distance of the shortest path between an initial and final position.

You use the concept of distance every day. For example, you are told a home run travels 400 ft (122 meters) or you run the metric mile (1.5 km) in track (or happily watch others run a metric mile).

Displacement adds the concept of direction to distance. For example, you go approximately 954 mi (1540 km) south when you travel from Seattle to Los Angeles; the summit of Mount Everest is 29,035 ft (8849.9 meters) above sea level. (You may have noticed we are using both metric and English units. We will do this only for the first part of this chapter, with the thought that this may prove helpful if you are familiarizing yourself with the metric system.)

Sometimes just distance matters. If you want to be a million miles away from your younger brother, it does not matter whether that’s east, north, west or south. The distance is called the magnitude − the amount − of the displacement.

Direction, however, can matter. If you walk 10 blocks north of your home, you are at a different location than if you walk 10 blocks south. In physics, direction often matters. For example, to get a ball to the ground from the top of a tall building, you can simply drop the ball. Throwing the ball back up requires a very strong arm. Both the direction and distance of the ball’s movement matters.

The definition of displacement is precise: the direction and length of the shortest path from the initial to the final position of an object’s motion. As you may recall from your mathematics courses, the shortest path between two points is a straight line. Physicists use arrows to indicate the direction of displacement. In the illustrations to the right, the arrow points in the direction of the mouse’s displacement.

Physicists use the Greek letter Δ (delta) to indicate a change or difference. A change in position is displacement, and since x represents position, we write Δx to indicate displacement. You see this notation, and the equation for calculating displacement, to the right. In the equation, x_f represents the final position (the subscript f stands for final) and x_i represents the initial position (the subscript i stands for initial).

Displacement is a vector. A vector is a quantity that must be stated in terms of its direction and its magnitude. Magnitude means the size or amount. “Move five meters to the right” is a description of a vector. Scalars, on the other hand, are quantities that are stated solely in terms of magnitude, like “a dozen eggs.” There is no direction for a quantity of eggs, just an amount.

In one dimension, a positive or negative sign is enough to specify a direction. As mentioned, numbers to the right of the origin are positive, and those to the left are negative. This means displacement to the right is positive, and to the left it is negative. For instance, you can see in Example 1 that the mouse’s car starts at the position +3.0 meters and moves to the left to the position −1.0 meters. (We measure the position at the middle of the car.) Since it moves to the left 4.0 meters, its displacement is −4.0 meters.

Displacement measures the distance solely between the beginning and end of motion. We can use dance to illustrate this point. Let’s say you are dancing and you take three steps forward and two steps back. Although you moved a total of five steps, your displacement after this maneuver is one step forward.

It would be better to use signs to describe the dance directions, so we could describe forward as “positive” and backwards as “negative.” Three steps forward and two steps back yield a displacement of positive one step.

Since displacement is in part a measure of distance, it is measured with units of length. Meters are the SI unit for displacement.