# An object travels North at 9 m/s for 2 s and then travels South at 7 m/s for  6 s. What are the object's average speed and velocity?

Jun 18, 2017

Average speed: $7.5 \text{m"/"s}$

Average velocity: $- 3 \text{m"/"s}$

#### Explanation:

We're asked to find an object's average speed and average velocity with a given displacement.

Let's find the average velocity first:

The object is moving in one dimension (which since it's north and south, we'll call the $y$-axis), so this makes things easy. We'll call north the positive $y$-direction, and south the negative $y$-direction.

The formula for the average velocity (along the $y$-axis in this case) is

${v}_{\text{av-y}} = \frac{\Delta y}{\Delta t}$

where

• $\Delta y$ is the total change in $y$-position, and

• $\Delta t$ is the change in time, which is $2$ $\text{s}$ $+ 6$ "s" = color(red)(8 color(red)("s"

We therefore need to find the total change in position, $\Delta y$.

For the first displacement (north), the distance traveled is

(9"m"/(cancel("s")))(2color(white)(l)cancel("s")) = 18 $\text{m}$

And the second displacement (south) is

(-7"m"/(cancel("s")))(6color(white)(l)cancel("s")) = -42 $\text{m}$

(negative because we're calling south the negative direction.)

The net displacement is

$18$ $\text{m}$ $+ \text{(} - 42$ $\text{m"")}$ = color(green)(-24 color(green)("m"

The average velocity is thus

${v}_{\text{av-y" = (Deltay)/(Deltat) = (color(green)(-24)color(white)(l)color(green)("m"))/(color(red)(8)color(white)(l)color(red)("s")) = color(blue)(-3"m"/"s}}$

Now, let's find the average speed:

The average speed of an object is

overbrace(v_"av-y")^"speed" = "total distance traveled"/(Deltat)

The total distance traveled is the sum of the two displacements, disregarding the signs.

It traveled $18$ meters north, and then $42$ meters south, so the total distance traveled is

$18$ $\text{m}$ $+ 42$ $\text{m}$ = color(purple)(60 color(purple)("m"

The time interval is the same, so the average speed is

overbrace(v_"av-y")^"speed" = "total distance traveled"/(Deltat) = (color(purple)(60)color(white)(l)color(purple)("m"))/(color(red)(8)color(white)(l)color(red)("s")) = color(darkorange)(7.5"m"/"s"

(It's always helpful to know that average speed is always a positive value, since we're only dealing with the magnitudes of the displacements, not the directions/signs.)