Signs and coordinate choices
You and a friend stand 10 m apart. Your dog, initially midway between you, runs in a straight line toward your friend. Each of you independently defines a coordinate system (position of origin and positive direction for the coordinate x) and calculates the dog’s average x component of velocity. (Remember that, in this context, when we say “velocity,” we really mean “x component of velocity.”) What can you say with certainty about the average velocity?
A. The average velocity you calculate is positive.
B. The average velocity your friend calculates is negative.
C. Neither of you can obtain a negative velocity.
D. Both of you can obtain a negative velocity
Let’s draw a simple sketch, like Figure 2.5, that shows you, your dog, and your friend. In our sketch, the dog runs to the right as it runs toward your friend. So the displacement of the dog points to the right, regardless of the coordinate system we choose. Because the average velocity vector points in the same direction as the displacement, the average velocity must also point to the right. This we can say with certainty, regardless of the coordinate system we choose. The sign of the average velocity, however, depends on the coordinate system we choose along the line joining you and your friend. Below the sketch, we have drawn two possible coordinate systems that we could choose for this problem. If we choose to the right to be the positive x direction, then the sign of the average velocity will be positive. If we let the positive x direction be to the left, then the sign of the average velocity will be negative. Since both you and your friend can pick the positive direction to point to either the left or the right, both of you can obtain an average velocity that is either positive or negative. Thus, the correct answer is D.
Notice that the sign of the average velocity does not depend on where you place the origin, but only on the direction you define as positive.