A shopper pushing a cart through a store
moves 30.0 m south down one aisle, then
makes a 90◦
turn and moves 11.0 m. He
then makes another 90◦
turn and moves 22.0
m.
What is the magnitude of the smallest possible
displacement the shopper could have?
You are not given the direction moved after
any of the 90◦
turns, so there could be more
than one answer.
Answer in units of m.
19 m
To find the magnitude of the smallest possible displacement, we need to consider the directions of the movements made by the shopper.
Let's break down the shopper's movements:
1. The shopper moves 30.0 m south down one aisle.
2. The shopper makes a 90° turn and moves 11.0 m.
3. The shopper makes another 90° turn and moves 22.0 m.
Since the question mentions that the directions after the 90° turns are not given, it means we have to consider all possible directions after each turn.
So let's explore the possible scenarios:
Scenario 1: After the first 90° turn, the shopper continues moving south.
In this case, the shopper moves an additional 11.0 m south, resulting in a displacement of 30.0 m + 11.0 m = 41.0 m.
Scenario 2: After the first 90° turn, the shopper moves north (opposite to the initial direction).
In this case, the shopper moves back 11.0 m north, canceling out the initial 30.0 m south displacement. The resulting displacement is 30.0 m - 11.0 m = 19.0 m.
Scenario 3: After the first 90° turn, the shopper moves east (perpendicular to the initial direction).
In this case, the shopper moves 11.0 m east, and the southward displacement of 30.0 m does not change. The resulting displacement is the diagonal of a right-angled triangle with sides 30.0 m and 11.0 m.
Using the Pythagorean theorem, the magnitude of the resulting displacement is √(30.0^2 + 11.0^2) = √(900 + 121) = √1021 ≈ 31.96 m.
Scenario 4: After the first 90° turn, the shopper moves west (opposite to perpendicular direction).
In this case, the shopper moves 11.0 m west, reducing the effective southward displacement to 30.0 m - 11.0 m = 19.0 m.
From the scenarios above, we find that the smallest possible displacement is 19.0 m.
So, the magnitude of the smallest possible displacement the shopper could have is 19.0 m.