The bacterium Escherichia coli (or E. coli ) is a...

The bacterium Escherichia coli (or $E$. coli ) is a singlecelled organism that lives in the gut of healthy humans and animals. When grown in a uniform medium rich in salts and amino acids, these bacteria swim along zig-zag paths at a constant speed of $20 \mu \mathrm{m} / \mathrm{s}$. Figure $\mathrm{P} 3.52$ shows the trajectory of an $E .$ coli as it moves from point A to point E. Each segment of the motion can be identified by two letters, such as segment $B C$ a. For each of the four segments in the bacterium's trajectory, calculate the $x$ - and $y$ -components of its displacement and of its velocity. b. Calculate both the total distance traveled and the magnitude of the net displacement for the entire motion. c. What are the magnitude and the direction of the bacterium's average velocity for the entire trip?

The bacterium Escherichia coli (or $E$. coli ) is a singlecelled organism that lives in the gut of healthy humans and animals. When grown in a uniform medium rich in salts and amino acids, these bacteria swim along zig-zag paths at a constant speed of $20 \mu \mathrm{m} / \mathrm{s}$. Figure $\mathrm{P} 3.52$ shows the trajectory of an $E .$ coli as it moves from point A to point E. Each segment of the motion can be identified by two letters, such as segment $B C$
a. For each of the four segments in the bacterium's trajectory, calculate the $x$ - and $y$ -components of its displacement and of its velocity.
b. Calculate both the total distance traveled and the magnitude of the net displacement for the entire motion.
c. What are the magnitude and the direction of the bacterium's average velocity for the entire trip?

Key Concepts

Vector Decomposition in Trajectory Analysis

This concept involves breaking down a complex motion into simpler segments that can be analyzed individually. For each segment or straight-line motion, one calculates the displacement and velocity components. After analysis, vector addition methods are applied to each component to understand the total effect of the motion. This framework is crucial in problems where motion is piecewise uniform or follows a zig-zag path.

Total Distance vs. Net Displacement

Total distance refers to the entire length of the path traveled regardless of the direction, while net displacement is the vector difference between the final and initial positions. In many problems, the path taken by an object versus its starting and ending points provides different insights: total distance measures how much ground has been covered, whereas net displacement conveys the overall effect of the movement in terms of position change.

Average Velocity

Average velocity is defined as the net displacement divided by the total time of the motion. It is a vector quantity that shows the overall direction and rate of change in position over the entire journey, and it may differ substantially from the simple average of speeds along the path if the movement involves turns or changes in direction.

Displacement Components

This concept refers to breaking a displacement vector into its x- and y- components. In any motion, especially in two-dimensional trajectories, the overall change in position can be represented by amounts in the horizontal (x) and vertical (y) directions. Calculating these components is essential for understanding and analyzing motions that are not aligned with the coordinate axes and for applying vector operations such as addition and subtraction.

Velocity Components

Velocity is a vector quantity characterized by both magnitude and direction. When given a constant speed along a specific path, its velocity can be broken down into x- and y- components using trigonometric functions based on the path’s orientation. This decomposition is important to relate the directional aspects of motion with the time taken for each segment, enabling accurate analysis of the motion’s dynamics along each coordinate axis.

Transcript

00:01 Hello and welcome to this video solution of numerate.

00:05 So here we are given that the bacterium equally is a single -celled organism that lives in gut of healthy human and animals.

00:18 When grown in a uniform medium rich in salt and amino acids, this bacteria swim along zigzag path at constant speed take it at u to be equal to 20 micrometer per second.

00:33 And this figure shows the trajectory of a e.

00:38 Coli as it moves from the point a to point e.

00:40 Each segment of the motion can be identified by two letters such as bc.

00:47 And based on each of the four segments in the bacterium's trajectory, you have to calculate the x and y component of the displacement and of its velocity.

01:02 So what we can do is we can simply copy this graph.

01:10 I'm drawing a simplified version let's say.

01:15 And sort of like this it moved straight up and then again the same path followed by a complete retracement back right.

01:28 So i will only mark the important points.

01:31 So this point is b.

01:34 This is a.

01:35 Start from a.

01:36 This is b which has got the position.

01:41 This is in micrometer.

01:43 And this is y x and y coordinates.

01:46 This is also in micrometers right.

01:49 B is a 50 micrometer and along the y direction it's 10 right.

01:58 And next you've got c the same 50 and that means it has moved only along the y direction now it's 20.

02:06 And it's d is 90 and 30 right.

02:12 This is 90 and this is 30 right.

02:18 And this e i can take it here.

02:20 This is minus 20 and this is 40 right.

02:26 This is the point e right.

02:28 Now based on this you have to calculate the x and y components of the displacement right.

02:41 Okay great.

02:42 So here we can clearly say that d x or maybe i segregate each segment for a b let's say.

02:54 So d x is the x component which is clearly 50 micrometer right.

03:00 And d y is the y component that is 10 micrometer.

03:08 Similarly for b c you have got d x to be equal to 0 and d y equal to you have got from 20 to 10 sorry from 10 to 20 which is another 10 right.

03:27 Or we can write 20 minus 10 mu m that's equal to 10 micrometer right.

03:33 Now from c d the x movement d x i can directly check it from the graph from 50 to 90 right.

03:51 So 40 mu m and d y will be in the vertical direction if you see here from 20 to 30 so 10 right.

04:03 And finally from d e for d e you have got coming from 30 to minus 20 sorry sorry in the x coordinate d x.

04:15 You can write like initially it was at 30 sorry it's 90 x position is 90 minus of the final position is 40 right.

04:31 Or yeah but you can do on the other way around because final minus initial we calculate.

04:41 So 40 minus of 90 right...

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