A delivery truck travels 19 blocks north, 12 blocks west and 22 blocks south. What is its final displacement from the origin? Assume the blocks are equal length. Find magnitude and direction. which is counter clockwise from east is positive.
To find the displacement from the origin, we can visualize the movement of the delivery truck on a coordinate grid.
First, let's assign positive as counterclockwise from east, which means clockwise from east would be negative. North is represented on the y-axis, and west/south are represented on the x-axis. Let's assume each block is 1 unit.
The truck traveled 19 blocks north, which means it moved up 19 units on the y-axis. This gives us a y-coordinate of +19.
Then, the truck traveled 12 blocks west, which means it moved 12 units to the left on the x-axis. This gives us an x-coordinate of -12.
Lastly, the truck traveled 22 blocks south, which means it moved down 22 units on the y-axis. This brings the y-coordinate to -3 (19 - 22 = -3).
Now, we can find the final displacement by using the coordinates (-12, -3). To calculate the magnitude, we use the Pythagorean theorem:
Magnitude = sqrt((-12)^2 + (-3)^2) = sqrt(144 + 9) = sqrt(153) ≈ 12.37 units
To determine the direction, we need to find the angle θ between the displacement vector and the positive x-axis.
θ = atan((-3)/(-12)) = atan(0.25) ≈ 14.04 degrees
Since we've defined counterclockwise from east as positive, the direction is 360 degrees - 14.04 degrees = 345.96 degrees counterclockwise from east.
Therefore, the final displacement from the origin is approximately 12.37 units in magnitude, with a direction of 345.96 degrees counterclockwise from east.