A - A girl delivering newspapers covers her route by traveling 4 blocks west, 2 blocks north, then 6 blocks east. What is her resultant displacement? Answer in units of blocks.
B - What is her direction? Use counterclockwise as the positive angular direction from due east, between the limits of −180◦ and 180◦. Answer in units of ◦.
C - What is her total distance? Answer in units of blocks.
went total of 6-4 = 2 blocks east (x axis)
went 2 blocks north (y axis)
net is sqrt (2^2+2^2) = 2 sqrt 2 northeast which is 45 degrees north of east
To solve this problem, let's break it down step by step:
A - To find the girl's resultant displacement, we need to calculate the net distance and direction from her starting point to her ending point.
First, let's calculate the displacement in the x-direction (east-west). The girl travels 4 blocks west and 6 blocks east. Therefore, her net displacement in the x-direction is 6 - 4 = 2 blocks east.
Next, let's calculate the displacement in the y-direction (north-south). The girl travels 2 blocks north. Therefore, her displacement in the y-direction is 2 blocks north.
Combining the displacements in the x and y-directions, we find that her resultant displacement is (2 blocks east, 2 blocks north).
B - To find her direction, we can use trigonometry. Let's define east as 0 degrees and counterclockwise as the positive angular direction.
We have a right-angled triangle where the horizontal leg represents the x-displacement (2 blocks east) and the vertical leg represents the y-displacement (2 blocks north).
Using the inverse tangent function, we can calculate the angle:
angle = arctan(y-displacement / x-displacement)
= arctan(2 / 2)
= arctan(1)
= 45 degrees
Since the angle is positive and counterclockwise, her direction is 45 degrees.
C - To find her total distance, we need to calculate the sum of the distances traveled in each direction.
The girl travels 4 blocks west, 2 blocks north, and 6 blocks east.
Total distance = |4 blocks west| + |2 blocks north| + |6 blocks east|
= 4 + 2 + 6
= 12 blocks
Therefore, her total distance is 12 blocks.
To find the resultant displacement, direction, and total distance of the girl delivering newspapers, we can break down her route into individual movements.
A - Resultant Displacement:
The girl travels 4 blocks west, 2 blocks north, then 6 blocks east. We need to calculate the difference between her total eastward movement and westward movement, as well as the difference between her total northward movement and southward movement.
Total eastward movement: 6 blocks
Total westward movement: 4 blocks
Resultant eastward movement: 6 - 4 = 2 blocks
Total northward movement: 2 blocks
Total southward movement: 0 (since she doesn't move south)
Resultant northward movement: 2 blocks
Now, we can use these resultant eastward and northward movements to calculate the resultant displacement. Using the Pythagorean theorem, the resultant displacement is the square root of the sum of the squares of these movements.
Resultant displacement = √(resultant eastward movement^2 + resultant northward movement^2)
Resultant displacement = √(2^2 + 2^2)
Resultant displacement = √(4 + 4)
Resultant displacement = √8
Resultant displacement ≈ 2.83 blocks
B - Direction:
To find the direction, we can use trigonometry. The direction is the angle formed between the resultant displacement and the due east direction (positive x-axis). We can use the cosine function to calculate this angle.
Direction = arccos(resultant eastward movement / resultant displacement)
Direction = arccos(2 / 2.83)
Direction ≈ arccos(0.71)
Direction ≈ 45.6°
Since the positive angular direction is counterclockwise from due east, the direction is approximately 45.6° counterclockwise from due east.
C - Total Distance:
To calculate the total distance, we need to add up the individual distances traveled in each movement.
Total distance = northward distance + eastward distance + westward distance
Total distance = 2 + 6 + 4
Total distance = 12 blocks
Therefore, the girl's resultant displacement is approximately 2.83 blocks, her direction is approximately 45.6° counterclockwise from due east, and her total distance is 12 blocks.
A. Disp. = -4 + 2i + 6 = 2 + 2i.
Disp. = sqrt(2^2+2^2) = 2.83 Blocks.
B. Tan A = 2/2 = 1. A = 45o N. of E.
C. Distance = 4 + 2 + 6 = 12 Blocks.