Distance and Displacement

How far ...?

Would you dance for food? Read how honeybees dance and use vectors to communicate the location of food.

Distance and displacement are two quantities which in common conversation are often used interchangeably. Nevertheless, the two are quite different and the distinction, though perhaps elusive at first, should become obvious from the following example. Suppose, for instance, a resident of Los Angeles decides to travel to San Francisco by way of New York. The distance covered will be about 6,000 miles, while the displacement will be about 400 miles roughly due North. Thus, distance tells us the total number of miles the Angelino has traveled, as registered, for instance, on an odometer. Displacement, on the other hand, indicates how far the Angelino actually got, irrespective of path. In addition, unlike distance, displacement contains a second piece of information, that is, a direction (in this example, North).

As you are pondering the distinction between distance and displacement, let us introduce two more handy definitions: scalar and vector. Scalars are objects which are fully described by a magnitude alone; examples include distance, speed, and the price of tea in China. Vectors are objects which are identified by both magnitude and direction; examples include displacement, velocity, and force. With these definitions, distance can be formally defined as the length of the path from A to B, where A and B represent the initial and final points. Displacement is a vector directed from A to B, with a magnitude given by the length of the straight line connecting A and B.

Example: Suppose Rex Tangle walks 4 m East, 2 m North, and finally 4 m West. What distance did he cover and what is his net displacement?

Solution: The distance is 4+2+4=10 m, while the displacement is 2 m North.

Example: Yute Erne is an expectant father who is pacing back and forth, at an average speed of 1.25 m/s inside a 20-m waiting area. What is Yute’s net displacement and what total distance does he cover after 100 trips past the receptionist?

Solution: The total distance covered during 50 round trips is 100·20=2,000 m, while the net displacement is zero. Yute is going nowhere fast!

Q: Three dogs are fighting over a bone. Dogs are pulling with equal force and making an angle of 120^o with each other. What is the outcome?

A: Assuming bone is not broken into pieces and none of the dogs let go of the bone, the bone will remain motionless, and no body wins. Show this using a vector diagram.

By now, you have noticed that vectors are not added and subtracted like pure numbers. You need to use Pythagorean triplets for many instances involving vector algebra. This is an interesting demo describing the rules of adding, subtracting and multiplying vectors. After familiarizing with the demo, show graphically Rex's net displacement.

Further Readings:

The Waggle dance (Would you dance for food?)

A Model of Non-cooperation (A tug of war among many)