A shopper pushes a cart 43 m south down one aisle and then turns 90.0° west and moves 25 m. He then makes another 90.0° turn and moves 14 m.
(a) What is the largest possible displacement of the shopper?
________m _________° counterclockwise from west
(b) What is the smallest possible displacement?
__________ m __________° counterclockwise from west
To find the largest and smallest possible displacements of the shopper, we need to calculate the vector sum of the individual displacements.
(a) Largest Possible Displacement:
To find the largest possible displacement, we consider the shopper's path as a right-angled triangle with two sides. The first side is the initial southward displacement of 43 m, and the second side is the combined displacement after the two 90° turns.
The second side of the triangle, after the first 90° turn, is the vector sum of moving westward 25 m and moving northward 14 m. We can find the magnitude of this vector sum using the Pythagorean theorem:
√(25^2 + 14^2) = √(625 + 196) = √821 ≈ 28.67 m
The largest possible displacement is the hypotenuse of the right-angled triangle, which is the vector sum of 43 m southward and 28.67 m counterclockwise from west. To calculate the angle counterclockwise from west, we can use the inverse tangent function:
angle = arctan(28.67/43) ≈ 36.66°
Therefore, the largest possible displacement of the shopper is approximately 43 m, 36.66° counterclockwise from west.
(b) Smallest Possible Displacement:
To find the smallest possible displacement, we need to consider the shopper's path as a vector triangle again. The first two displacements remain the same (43 m south and then 25 m west).
Now, the final displacement of 14 m can be subtracted from the shopper's total displacement. Since we are subtracting, we assume the direction opposite to the original displacement of 14 m. Therefore, the smallest possible displacement is the vector sum of 43 m southward and 25 m westward minus 14 m eastward.
Subtracting the displacement of 14 m eastward from the vector sum of the first two displacements:
√((43^2 + 25^2) - 14^2) = √(1849 + 625 - 196) = √2278 ≈ 47.75 m
To find the angle counterclockwise from west, we can again use the inverse tangent function:
angle = arctan(47.75/25) ≈ 62.47°
Therefore, the smallest possible displacement of the shopper is approximately 47.75 m, 62.47° counterclockwise from west.