Question

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?

Answer 1

Average speed: `7.5"m"/"s"`

Average velocity: `-3"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_"av-y" = (Deltay)/(Deltat)`

where

• `Deltay` is the total change in `y`-position, and

• `Deltat` is the change in time, which is `2` `"s"` `+ 6` `"s" = color(red)(8` `color(red)("s"`

We therefore need to find the total change in position, `Deltay`.

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

`(9"m"/(cancel("s")))(2color(white)(l)cancel("s")) = 18` `"m"`

And the second displacement (south) is

`(-7"m"/(cancel("s")))(6color(white)(l)cancel("s")) = -42` `"m"`

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

The net displacement is

`18` `"m"` `+ "("-42` `"m"")"` `= color(green)(-24` `color(green)("m"`

The average velocity is thus

`v_"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` `"m"` `+ 42` `"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.)