An object travels North at  8 m/s for 8 s and then travels South at  5 m/s for  2 s. What are the object's average speed and velocity?

Jun 21, 2017

$\text{average speed" = 7.4"m"/"s}$

$\text{average velocity" = 5.4"m"/"s}$

Explanation:

We're asked to find both the average speed and the average velocity of the object with given displacements.

Let's calculate the average velocity first.

The average velocity of an object is defined by the equation

${\vec{v}}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t}$

But since this object is moving only northward and southward, which we'll say is along the $y$-axis, we can simplify this to the $y$-form:

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

where

• ${v}_{\text{av-y}}$ is the average $y$-velocity

• $\Delta y$ is the change in $y$-position of the object

• $\Delta t$ is the total change in time, which is $8$ $\text{s}$ $+ 2$ $\text{s}$ = color(blue)(10 color(blue)("s"

For reference, we'll call north the positive $y$-axis and south the negative $y$-axis. We'll first find out how far it traveled in each displacement:

1: (8"m"/(cancel("s")))(8cancel("s")) = 64 $\text{m}$ (north)

2: (-5"m"/(cancel("s")))(2cancel("s")) = -10 $\text{m}$ (south)

The total change in $y$-position is

$64$ $\text{m}$ + (-10color(white)(l)"m") = color(red)(54 color(red)("m"

And so the average velocity is

${v}_{\text{av-y" = (color(red)(54)color(white)(l)color(red)("m"))/(color(blue)(10)color(white)(l)color(blue)("s")) = color(purple)(5.4"m"/"s}}$

Now we'll find the average speed of the object.

The average speed of an object is defined as

overbrace(v_"av")^"speed" = "total distance travaled"/(Deltat)

The total distance traveled is merely the sum of the distances traveled disregarding the signs and directions, so the total distance is

$64$ $\text{m}$ $+ 10$ $\text{m}$ = color(green)(74 color(green)("m"

And the average speed is thus

$\overbrace{{v}_{\text{av")^"speed" = (color(green)(74)color(white)(l)color(green)("m"))/(color(blue)(10)color(white)(l)color(blue)("s")) = color(orange)(7.4"m"/"s}}}$