2.1 Displacement

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 $x_0=1\text{.}5\text{m}$ and her final position is $x_{\text{f}}=3\text{.}5\text{m}$ . 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.

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?

(a) The rider's displacement is $\mathrm{\Delta }x=x_{\mathrm{f}}-x_0=\text{−1 km}$ . (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 $\mathrm{1\; km}$ .

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Section summary

• Kinematics is the study of motion without considering its causes. In this chapter, it is limited to motion along a straight line, called one-dimensional motion.
• Displacement is the change in position of an object.
• In symbols, displacement $\mathrm{\Delta }x$ is defined to be
  $\mathrm{\Delta }x=x_{\mathrm{f}}-x_0,$
  where $x_0$ is the initial position and $x_{\mathrm{f}}$ is the final position. In this text, the Greek letter $\mathrm{\Delta }$ (delta) always means “change in” whatever quantity follows it. The SI unit for displacement is the meter (m). Displacement has a direction as well as a magnitude.
• When you start a problem, assign which direction will be positive.
• Distance is the magnitude of displacement between two positions.
• Distance traveled is the total length of the path traveled between two positions.

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