A shopper pushes a cart 38 m south down one aisle and then turns 90.0° west and moves 25 m. He then makes another 90.0° turn and moves 17 m.
(a) What is the largest possible displacement of the shopper?
m ° counterclockwise from west
(b) What is the smallest possible displacement of the shopper?
m ° counterclockwise from west
To find the displacement of the shopper, we need to determine the distance and direction the shopper travels in each step.
Let's break down the shopper's movements step by step.
Step 1: The shopper moves 38 m south. This means the shopper's distance traveled in this step is 38 m, and the direction is south.
Step 2: After moving south, the shopper makes a 90.0° west turn and moves 25 m. This means the shopper's distance traveled in this step is 25 m, and the direction is west.
Step 3: Finally, after moving west, the shopper makes another 90.0° turn and moves 17 m. This means the shopper's distance traveled in this step is 17 m, and the direction is north.
Now, let's calculate the displacement of the shopper:
(a) To find the largest possible displacement, we need to add the distances traveled in each direction. Since the shopper moves south (negative y-direction), west (negative x-direction), and later north (positive y-direction), the largest possible displacement occurs when the shopper is farthest north.
Step 1: -38 m (southward)
Step 2: -25 m (westward)
Step 3: +17 m (northward)
To get the displacement, we add the distances:
Displacement = -38 m + (-25 m) + 17 m
Calculating the sum, we find that the largest possible displacement is -46 m. The negative sign indicates that the displacement is in the south direction, and the magnitude of 46 m represents the distance.
Therefore, the largest possible displacement of the shopper is 46 m south.
(b) To find the smallest possible displacement, we need to add the distances traveled in each direction. Since the shopper moves south (negative y-direction), west (negative x-direction), and later north (positive y-direction), the smallest possible displacement occurs when the shopper is farthest south.
Step 1: -38 m (southward)
Step 2: -25 m (westward)
Step 3: +17 m (northward)
To get the displacement, we add the distances:
Displacement = -38 m + (-25 m) + 17 m
Calculating the sum, we find that the smallest possible displacement is -46 m. The negative sign indicates that the displacement is in the south direction, and the magnitude of 46 m represents the distance.
Therefore, the smallest possible displacement of the shopper is 46 m south.