Displacement

Position

In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See Figure 2.4.)

Displacement

If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object’s position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.

In this text the upper case Greek letter $\text{Δ}$ (delta) always means “change in” whatever quantity follows it; thus, $\text{Δ}x$ means change in position. Always solve for displacement by subtracting initial position $x_0$ from final position $x_{\text{f}}$.

Note that the SI unit for displacement is the meter (m) (see Physical Quantities and Units), but sometimes kilometers, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation.

Figure 2.3 A professor paces left and right while lecturing. Her position relative to Earth is given by $x$. The $+2\text{.}0\mathrm{<mspace\; width="0.25em"><mtext>m</mtext></mspace>}$ displacement of the professor relative to Earth is represented by an arrow pointing to the right.

Figure 2.4 A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by $x$. The $-4\text{.}0\text{-m}$ displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far) in Figure 2.3.

Note that displacement has a direction as well as a magnitude. The professor’s displacement is 2.0 m to the right, and the airline passenger’s displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction). The professor’s initial position is $x_0=1\text{.}5\mathrm{<mspace\; width="0.25em"><mtext>m</mtext></mspace>}$ and her final position is $x_{\text{f}}=3\text{.}5\mathrm{<mspace\; width="0.25em"><mtext>m</mtext></mspace>}$. Thus her displacement is

In this coordinate system, motion to the right is positive, whereas motion to the left is negative. Similarly, the airplane passenger’s initial position is $x_0=6\text{.}\mathrm{0\; m}$ and his final position is $x_{\mathrm{f}}=2\text{.}\mathrm{0\; m}$, so his displacement is

His displacement is negative because his motion is toward the rear of the plane, or in the negative $x$ direction in our coordinate system.

Distance

Although displacement is described in terms of direction, distance is not. Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m.

Check Your Understanding

A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement?

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Solution

Figure 2.5

(a) The rider’s displacement is $\mathrm{\Delta }x=x_{\text{f}}-x_0=\text{−1 km}$. (The displacement is negative because we take east to be positive and west to be negative.)

(b) The distance traveled is $\text{3 km}+\text{2 km}=\text{5 km}$.

(c) The magnitude of the displacement is $\mathrm{1\; km}$.