A shopper pushes a cart 50 m south down one aisle and then turns 90.0° west and moves 19 m. He then makes another 90.0° turn and moves 13 m.
(a) What is the largest possible displacement of the shopper?
(b) What is the smallest possible displacement of the shopper?
Could you show me step by step
start the shopper at (0,0)
After pushing the cart 50 m south, he's now at (0, -50)
After turning west and pushing the cart 19 m, he's now at (-19, -50)
Now he has to turn 90 degrees again. . . he will then be going either north or south. . . south will result in the largest displacement from the origin. . . now he's at (-19, -63)
north will result in the smallest displacement from the origin . . . then he'd be at (-19, -37)
The distance from the origin for the largest displacement is (19^2 + 63^2)^0.5
Let the angle counterclockwise from west be x: then
tan x = 63/19
The distance from the origin for the smallest displacement is (19^2 + 37^2)^0.5
at an angle:
tan x = 37/19
Sure! Let's break down the problem step by step:
(a) To find the largest possible displacement of the shopper, we need to consider that displacements are vectors and can be represented as arrows. The magnitude of the displacement is the distance between the starting and ending points, while the direction is the angle between the line connecting the starting and ending points and a reference direction (typically the positive x-axis).
Step 1: Determine the initial position. The shopper starts at the origin (0, 0).
Step 2: Calculate the first displacement. The shopper pushes the cart 50 m south, so the first displacement is a vector pointing 50 m directly south. This vector can be represented as D1 = (0, -50).
Step 3: Calculate the second displacement. The shopper turns 90° west and moves 19 m. To find the resulting displacement vector, we can use vector addition. Since the shopper moves west, the x-coordinate decreases by 19, and since there is no change in the y-coordinate, the y-coordinate remains the same. Therefore, the second displacement can be represented as D2 = (-19, -50).
Step 4: Calculate the third displacement. The shopper makes another 90° turn and moves 13 m. Again, using vector addition, we add this displacement to the previous displacement. Since the shopper moves north, the y-coordinate increases by 13, and since there is no change in the x-coordinate, the x-coordinate remains the same. Therefore, the third displacement can be represented as D3 = (-19, -50 + 13) = (-19, -37).
Step 5: Calculate the total displacement. To find the total displacement, we sum up all the individual displacements: D_total = D1 + D2 + D3 = (0, -50) + (-19, -50) + (-19, -37).
Let's calculate the total displacement:
x-coordinate = 0 - 19 - 19 = -38
y-coordinate = -50 - 50 - 37 = -137
Therefore, the total displacement vector is D_total = (-38, -137).
Now we can find the magnitude and direction:
Magnitude: The magnitude of the displacement vector is the distance between the initial and final positions of the shopper. We can use the Pythagorean theorem to calculate it:
|D_total| = sqrt((-38)^2 + (-137)^2) ≈ 141.63 m
Direction: The direction of the displacement can be calculated using trigonometry. The angle between the displacement vector and the positive x-axis can be found using the arctan function:
θ = arctan(-137 / -38) ≈ 73.57°.
Therefore, the largest possible displacement of the shopper is approximately 141.63 m at an angle of 73.57° counterclockwise from the positive x-axis.
(b) To find the smallest possible displacement, we need to consider that the shopper is returning to the starting point.
Step 6: Calculate the final displacement. The shopper needs to move back to the starting point, so the final displacement is the negative of the total displacement vector: D_final = -D_total = (-(-38), -(-137)) = (38, 137).
The magnitude and direction of the smallest possible displacement are the same as the largest possible displacement, but in the opposite direction. So, the smallest possible displacement of the shopper is approximately 141.63 m at an angle of 253.57° counterclockwise from the positive x-axis.