The bacterium Escherichia coli (or E. coli) is a single celled organism that lives in the gut of healthy humans and animals. When grown in a uniform medium rich in salts and amino acids, it swims along zigzag paths at a constant speed changing direction at varying time intervals. Figure P1.61 shows the positions of an E. coli as it moves from point A to point J. Each segment of the motion can be identified by two letters, such as segment BC. During which segments, if any, does the bacterium have the same a. Displacement? b. Speed? c. Velocity?
ANSWER: a) Same displacement The concept of displacement is, an object is displaced about how much amount and it must has a specific directions. This is the reason we call displacement as a vector quantity. As we know that distance is a scalar quantity, it can have any orientation. With this knowledge if we proceed, we can see from the above figure that AB, CD and IH have same magnitude. EF and FG have same magnitude. But the difference is, IH has a different orientation. EF and FG also have different orientations. So these segments are none of our business and the only segments we are left with are AB and CD. They both have same magnitude and same direction. So they have same displacement. b) Same speed As the instruction given, the bacterium only changes it’s direction not the speed. The speed is constant for all the segments. So all segments have equal speed. Because speed is a scalar, completely defined by the magnitude and it is independent of the direction. This implies that, AB, CD, DE, EF, FG, GH, HI, IJ all have same speed. c) Same velocity As we saw in a), only AB and CD segments have equal displacement, this gives complete information about the equal velocity. We all know that velocity is a vector quantity which is direction sensitive. It’s completely defined by it’s direction and magnitude. So in the above figure there is one and only one case where this condition satisfies, so for the segments AB and CD the velocity is equal.