A whiny toddler is in the exact center of a 20-foot long room. At one end of the room is a lollipop and at the other end is a teddy bear. The toddler toddles back and forth with velocity given by the graph below. When velocity is positive, it means the toddler is moving towards the teddy bear.

(a) Describe the toddler's location at *t* = 2, 4, 6, and 10 seconds.

(b) How many times between *t* = 0 and *t* = 10 seconds will the toddler pass through the exact center of the room?

(c) Will the toddler eventually reach the lollipop, the teddy bear, or neither?

Answer

First we need to understand the set-up. A toddler is between a lollipop and a teddy bear, each of which is 10 feet away from the toddler. We put the teddy bear on the right side because the problem says positive velocities mean the toddler is moving towards the teddy bear. We're used to thinking of positive velocities as moving us to the right on a number line, so if we put the teddy bear on the right side of the room, positive velocity still corresponds to moving right.

(a) We look at the graph to see how far the toddler moves over each time interval.

From *t* = 0 to *t* = 2 seconds the toddler moves

to the right, so the toddler is 14 feet from the lollipop and 6 feet from the teddy bear:

From *t* = 2 to *t* = 4 seconds the toddler moves

which means it moves 2 feet to the left (closer to the lollipop). The toddler is now 12 feet from the lollipop and 8 feet from the teddy bear.

From *t* = 4 to *t* = 6 seconds the toddler moves

closer to the bear, so the toddler is now 15 feet from the lollipop and 5 feet from the bear.

From *t* = 6 to *t* = 10 the toddler moves

meaning 6 feet to the left (closer to the lollipop). The toddler is now 9 feet from the lollipop and 11 feet from the bear.

What we did here was put the lollipop, toddler, and bear on a number line. The lollipop was at -10, the toddler at 0, and the bear at 10.

There are other ways to do this problem. For example,

we could put the lollipop at 0, the bear at 20, and start the toddler off at 10.

Then the toddler would end up at

10 + 4 – 2 + 3 – 6 = 9

on the numberline, meaning 9 feet from the lollipop and 11 feet from the bear.

Another possibility is that we could put the lollipop at -20, the bear at 0, and the toddler at -10:

Any way we do it, the toddler will end up 9 feet from the lollipop and 11 feet from the bear.

(b) The toddler is at the center of the room when *t* = 0, and passes through the center of the room once more between *t* = 6 and *t* = 10, in order to get closer to the lollipop than to the bear. This means the toddler passes through the center of the room twice.

(c) The toddler never reaches the lollipop or the bear. The toddler gets within 5 feet of the bear, and within 9 feet of the lollipop, but never gets to either.