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?

1 Answer
Jun 18, 2017

Answer:

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.)