Question Video: Comparing the Distance and Displacement of a Body Moving in a Given Path Mathematics

Video Transcript

According to the figure, a body moved from 𝐴 to 𝐵 along the line segment 𝐴𝐵, and then it moved to 𝐶 along 𝐵𝐶. Finally, it moved to 𝐷 along 𝐶𝐷 and stopped there. Find the distance covered by the body 𝑑 one and the magnitude of its displacement 𝑑 two.

Looking at the figure, we’re told that a body begins at point 𝐴, right here, and then follows the line segment 𝐴𝐵 to point 𝐵 then moves to point 𝐶 following this path, finally moving along line segment 𝐶𝐷 to end up at point 𝐷. Given this motion, we want to calculate the distance the body has covered, we call that 𝑑 one, and the magnitude of its displacement 𝑑 two.

Clearing some space, let’s first work on solving for the distance our body travels 𝑑 one. We can recall that distance in general is equal to the total path length followed by some body as it moves from one location to another. In our case, our body moved from point 𝐴 to point 𝐷 along the path shown in orange. We see that that involves 6.6 centimeters of travel from point 𝐴 to point 𝐵, 8.8 centimeters from 𝐵 to 𝐸, 16.4 centimeters from 𝐸 to 𝐶, and on the last leg of the journey 12.3 centimeters. 𝑑 one then equals the sum of these four distances. Adding them all up gives a result of 44.1 centimeters. This is the distance our body traveled.

And now let’s consider its displacement magnitude 𝑑 two. Displacement is different from distance in that displacement only takes into account the start point and end point for some body. In our situation, our body begins at point 𝐴 and it ends up at point 𝐷. So this straight pink line connecting these two points represents the displacement magnitude 𝑑 two. We see that the length of this line segment can be divided up into two parts: one part right here and then the second part here. Each of these is a hypotenuse of a right triangle. This means we can use the Pythagorean theorem to solve for these lengths.

If we call the length of the first hypotenuse 𝑙 one and that of the second 𝑙 two, then we can say that the displacement magnitude 𝑑 two is equal to their sum and that 𝑙 one and 𝑙 two are defined this way. 𝑙 one equals the square root of 6.6 centimeters quantity squared plus 8.8 centimeters quantity squared. And 𝑙 two equals the square root of 16.4 centimeters quantity squared plus 12.3 centimeters quantity squared. When we enter these expressions on our calculator, we find that 𝑙 one is equal to exactly 11 centimeters, while 𝑙 two is 20.5 centimeters. Adding these together gives us a result of 31.5 centimeters. This is the displacement magnitude of our body.