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, these bacteria swim along zig-zag paths at a constant speed of 20 um/s. The figure 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 BC.
a) for each of the four segments in the bacterium's trajectory, calculate the x and y components 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 and for the entire trip?
The left ventricle of the heart accelerates blood from rest to a velocity of +22 cm/s.
(a) If the displacement of the blood during the acceleration is +2.2 cm, determine its acceleration (in cm/s2).
cm/s2
(b) How much time does blood take to reach its final velocity?
s
a) To calculate the x and y components of the bacterium's velocity for each segment, we need to determine the change in position in the x and y directions divided by the time taken.
Segment AB:
Change in x = 20 um - 10 um = 10 um
Change in y = 10 um - 0 um = 10 um
Time taken = 5 s - 0 s = 5 s
Velocity components in segment AB:
Vx = (Change in x) / (Time taken) = 10 um / 5 s = 2 um/s
Vy = (Change in y) / (Time taken) = 10 um / 5 s = 2 um/s
Segment BC:
Change in x = 10 um - 20 um = -10 um
Change in y = 20 um - 10 um = 10 um
Time taken = 10 s - 5 s = 5 s
Velocity components in segment BC:
Vx = (Change in x) / (Time taken) = -10 um / 5 s = -2 um/s
Vy = (Change in y) / (Time taken) = 10 um / 5 s = 2 um/s
Segment CD:
Change in x = 30 um - 10 um = 20 um
Change in y = 20 um - 20 um = 0 um
Time taken = 20 s - 10 s = 10 s
Velocity components in segment CD:
Vx = (Change in x) / (Time taken) = 20 um / 10 s = 2 um/s
Vy = (Change in y) / (Time taken) = 0 um / 10 s = 0 um/s
Segment DE:
Change in x = 30 um - 30 um = 0 um
Change in y = 0 um - 10 um = -10 um
Time taken = 30 s - 20 s = 10 s
Velocity components in segment DE:
Vx = (Change in x) / (Time taken) = 0 um / 10 s = 0 um/s
Vy = (Change in y) / (Time taken) = -10 um / 10 s = -1 um/s
b) To calculate the total distance traveled, we can sum up the distances traveled in each segment.
Distance traveled in segment AB:
Distance = √[(Change in x)^2 + (Change in y)^2] = √[(10 um)^2 + (10 um)^2] = √[200 um^2] = 14.14 um
Distance traveled in segment BC:
Distance = √[(Change in x)^2 + (Change in y)^2] = √[(-10 um)^2 + (10 um)^2] = √[200 um^2] = 14.14 um
Distance traveled in segment CD:
Distance = √[(Change in x)^2 + (Change in y)^2] = √[(20 um)^2 + (0 um)^2] = √[400 um^2] = 20 um
Distance traveled in segment DE:
Distance = √[(Change in x)^2 + (Change in y)^2] = √[(0 um)^2 + (-10 um)^2] = √[100 um^2] = 10 um
Total distance traveled = 14.14 um + 14.14 um + 20 um + 10 um = 58.28 um
To calculate the net displacement, we need to determine the overall change in x and y positions.
Net displacement = Final position - Initial position
Change in x = 30 um - 10 um = 20 um
Change in y = 0 um - 0 um = 0 um
Magnitude of net displacement = √[(Change in x)^2 + (Change in y)^2] = √[(20 um)^2 + (0 um)^2] = √[400 um^2] = 20 um
c) The average velocity can be calculated by dividing the net displacement by the total time taken.
Total time taken = 30 s - 0 s = 30 s
Average velocity = Net displacement / Total time taken = 20 um / 30 s = 0.67 um/s
The magnitude of the average velocity is 0.67 um/s. Since the net displacement is in the positive x direction, the direction of average velocity is along the positive x-axis.
To solve this problem, we need to break it down into steps and calculate each component separately.
a) For each of the four segments in the bacterium's trajectory, we need to calculate the x and y components of its velocity.
- Segment AB: In this segment, we move from point A to point B. Since the bacterium is moving along a horizontal line, there is no change in the y-component of its velocity. The x-component is the distance traveled divided by the time taken. Let's assume the time taken is t seconds. The distance traveled is given by the equation: distance = speed * time. Therefore, the x-component of velocity is (distance / t).
- Segment BC: In this segment, we move from point B to point C. There is no change in the x-component of velocity because the bacterium is moving along a vertical line. The y-component can be calculated using the same method as explained for segment AB.
- Segment CD: In this segment, we move from point C to point D. Calculate the x and y components of velocity using the same method as explained for segments AB and BC.
- Segment DE: In this segment, we move from point D to point E. Calculate the x and y components of velocity using the same method as explained for segments AB, BC, and CD.
b) To calculate both the total distance traveled and the magnitude of the net displacement for the entire motion, we need to sum up the distances traveled and the displacements in the x and y directions.
- Total distance traveled: Sum up the distances traveled in each segment using the formula: total_distance = distance_AB + distance_BC + distance_CD + distance_DE.
- Magnitude of net displacement: To calculate the net displacement, we need to calculate the change in the x and y coordinates between the starting point (A) and the ending point (E). The net displacement is given by the formula: displacement = √((change_in_x)^2 + (change_in_y)^2).
c) To determine the magnitude and direction of the bacterium's average velocity for the entire trip, we need to calculate the change in x and y coordinates between points A and E, and divide it by the total time taken for the entire trip.
- Average velocity magnitude: Calculate the magnitude of average velocity using the formula: average_velocity_magnitude = total_distance / total_time.
- Average velocity direction: Determine the angle between the net displacement vector (change in x and y coordinates) and the positive x-axis using trigonometry or vector operations.
By following these steps and performing the necessary calculations, you can find the answers to all the sub-questions in this problem.