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An object travels North at # 5 m/s# for #3 s# and then travels South at # 4 m/s# for # 6 s#. What are the object's average speed and velocity?

Answer 1

Average speed: #4.33"m"/"s"#

Average velocity: #-1"m"/"s"#

For a given displacement, we are to determine the average speed as well as the average velocity.

First, let's figure out the average velocity.

The average velocity of an object moving in one dimension (which we'll call #y#, as it's moving north and south) is defined as

#v_(av-y) = (Deltay)/(Deltat)#

where

#Deltay# is the total change in the object's position (what we must find), and

#Deltat# is the total change in time, which is

#overbrace(3"s")^"first displacement" + overbrace(6"s")^"second displacement" = overbrace(9"s")^"total"#

We will designate north as "positive" and south as "negative" in order to determine the object's total net change in position after both displacements. The two displacements are

#1#:

#5"m"/(cancel("s"))(3cancel("s")) = 15# #"m"#

#2#:

#-4"m"/(cancel("s"))(6cancel("s")) = -24# #"m"#

The sum of the individual displacements determines the total displacement:

#15# #"m" + (-24# #"m") = -9# #"m"#

Consequently, the object's average velocity is

#v_(av-y) = (Deltay)/(Deltat) = (-9"m")/(9"s") = color(red)(-1)# #color(red)("m"/"s")#

Now, let's calculate the average speed of the object, which is slightly different than the average velocity. The average speed is the total distance traveled divided by the time interval (still #9# #"s"#):

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

To find the total distance traveled, we neglect all signs when it comes to the object's position. Thus, traveling #15# #"m"# north and then #24# #"m"# south is a total of

#15# #"m" + 24# #"m" = 39# #"m"#

The object's average speed is therefore

#overbrace(v_(av-y))^"speed" = ("total distance traveled")/(Deltat) = (39"m")/(9"s") = color(blue)(4.33"m"/"s"#

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Answer 2

Isabella Alvarez

The object's average speed is ( \frac{total\ distance}{total\ time} = \frac{(5\ m/s \times 3\ s) + (4\ m/s \times 6\ s)}{3\ s + 6\ s} = \frac{(15\ m) + (24\ m)}{9\ s} = \frac{39\ m}{9\ s} = 4.33\ m/s ).

To find the average velocity, we consider the displacement. The object travels north for 15 meters and south for 24 meters, resulting in a net displacement of 9 meters south. Therefore, the average velocity is ( \frac{displacement}{total\ time} = \frac{9\ m}{9\ s} = 1\ m/s ) south.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.