(a) A hotdog vendor named Sam is walking from the southwest corner of Central park to the Empire State Building. He starts at the intersection of 59th Street and Eighth Avenue, walks 3 blocks east to 59th Street and Fifth Avenue, and then 25 blocks south to the Empire State Building at the Corner of 34th Street and Fifth Avenue. Write his displacement vector in rectangular notation with units of "blocks." Orient the axes so positive y is to the north, and positive x is to the east. (b) A stockbroker named Andrea makes the same trip in a cab that gets lost, and detours 50 blocks south to Washington Square before reorienting and finally arriving at the Empire State Building. Is her displacement any different?
a. Draw a straight hor. arrow pointing to the right. From the head of this arrow, draw a ver. arrow pointing downward to form a right angle. Draw the hypotenuse to form a right triangle. The length of the hyp. is the displacement from the starting point: Disp. = 3 - 25i. Magnitude = sqrt(3^2+25^2)
b. No. The displacement is the straight-line distance from starting point to destination which is the same in both cases.
(a) To find the displacement vector of Sam, we need to determine the change in position from his starting point to the destination point.
First, let's establish a coordinate system with the southwest corner of Central Park as the origin (0, 0). We orient the axes so that the positive y direction is north and the positive x direction is east.
Starting at the intersection of 59th Street and Eighth Avenue (block A), Sam walks 3 blocks east along 59th Street to the intersection of 59th Street and Fifth Avenue (block B).
The change in x-coordinate from A to B is +3 blocks (since he moved to the east).
Next, he walks 25 blocks south along Fifth Avenue to reach the Empire State Building at the corner of 34th Street and Fifth Avenue (block C).
The change in y-coordinate from B to C is -25 blocks (since he moved to the south).
Therefore, the displacement vector from A to C is (Δx, Δy) = (+3 blocks, -25 blocks).
In rectangular notation, the displacement vector is (3, -25) blocks.
(b) Andrea's displacement will be different because she made a detour to Washington Square before reaching the Empire State Building.
Let's assume Andrea starts at the same point as Sam (intersection of 59th Street and Eighth Avenue) and goes on the detour, moving 50 blocks south to Washington Square.
The change in y-coordinate from A to Washington Square is -50 blocks.
Next, Andrea heads back towards the Empire State Building and reaches the corner of 34th Street and Fifth Avenue.
The change in x-coordinate from Washington Square to the Empire State Building is 25 blocks (since she moved to the east).
Therefore, the displacement vector for Andrea from A to the Empire State Building is (Δx, Δy) = (+25 blocks, -50 blocks).
In rectangular notation, the displacement vector is (25, -50) blocks.
Comparing the two displacement vectors, we can see that Andrea's displacement vector is different from Sam's due to the detour she took.