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Displacement

$\Delta \vec{x}\equiv\vec x_f- \vec x_i\,$

Displacement answers the question, "Has the object moved?"

Note the $\equiv$ symbol. This symbol is a sort of "super equals" symbol, indicating that not only does $\vec x_f- \vec x_i$ EQUAL the displacement $\Delta\vec{x}$ , but more importantly displacement is OPERATIONALLY DEFINED by $\vec x_f- \vec x_i$ .

We say that $\vec x_f- \vec x_i$ operationally defines displacement, because $\vec x_f- \vec x_i$ gives a step by step procedure for determining displacement. Namely ...

Measure where the object is initially.
Measure where the object is at some later time.
Determine the difference of these two position values.

Be sure to note that DISPLACEMENT is NOT the same as DISTANCE traveled.

For example, imagine traveling one time along the circumference of a circle. If you end where you started, your displacement is zero, even though you have clearly traveled some distance. In fact, displacement is an average distance traveled. On your trip along the circle, your north and south motion averaged out, as did your east and west motion.

Clearly we are losing some important information. The key to regaining this information is to use smaller displacement intervals. For example, instead of calculating your displacement for your trip along the circle in one large step, consider dividing the circle into 16 equal segments. Calculate the distance you traveled along each of these segments, and then add all your results together. Now your total traveled distance is not zero, but something approximating the circumference of the circle. Is your approximation good enough? Ultimately, that depends on the level of accuracy you need in a particular application, but luckily you can always use finer resolution. For example, we could break your trip into 32 equal segments for a better approximation.

Returning to your trip around the circle, you know the true distance is simply the circumference of the circle. The problem is that we often face a practical limitation for determining the true distance traveled. (The traveled path may have too many twists and turns, for example.) Luckily, we can always determine displacement, and by carefully choosing small enough displacement steps, we can use displacement to obtain a pretty good approximation for the true distance traveled. (The mathematics of calculus provides a formal methodology for formally estimating a "true value" through the use of successively better approximations.) In the rest of this discussion, I will replace ? with ? to indicate that small enough displacement steps have been used to provide a good enough approximation for the true distance traveled.