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In this problem you will try to make sense of how we use signs to specify direction in our specification of motion. Consider a small cart confined to move along a one-dimensional track as shown in the figure below. We specify the position of the cart by using the x-coordinate shown. The position of the cart is taken to be specified by the value of x that the little arrow on the front of the cart points to.

We specify the cart's position along the track by using a vector

.

The i-hat indicates the positive x direction, while the coordinate x specifies the distance from the origin. If it is positive it is on the right of the origin, if negative it is on the left.

We describe our motion in terms of position, velocity, and acceleration. Each can independently be positive, negative, or zero.

(a) Define velocity and acceleration for the cart in terms of our vector notation.

(b) At any instant of time,
there are 27 different combinations (!) of +/-/0 for position, velocity,
and acceleration (= 3 x 3 x 3) . Are all of these possible? For each case
that is possible, give a brief description of where the car is and how it
is moving, and sketch a bit of the graphs for position, velocity, and acceleration
vs. time. For each case that is not possible, give an explanation of why
it isn't. (*Hint*:
Paying attention to the patterns that develop may speed up the task.) As
an example, note that for the case of +/+/+ the car is on the positive axis,
is moving in the positive direction, and is speeding up. The position, velocity,
and acceleration graphs should look like the figures shown at the left below.
For the case of +/-/0, the car is on the positive axis, is moving in the
negative direction at a constant velocity (0 acceleration) so the graphs
would look like the figures on the right below.

+/+/+ |
+/-/0 |

(Problem suggested by Dan Campbell)

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