A man in search of his dog drives first 13.3 mi northeast, then 17.3 mi straight south, and finally 6.85 mi in a direction 30.3° north of west.
a) What is the magnitude of his resultant displacement?
b) What is the direction of his resultant displacement? Express your answer counterclockwise relative to the positive x-axis.
Here are the formulas that'll lead you to the answer. Just do the math to solve for the horizontal and vertical, then plug and chug.
A)
Horizontal = 13.3cos45 + 0 - 6.85cos30.3
Vertical = 13.3sin45 - 17.3 + 6.85sin30.3
(Vertical)^2 + (Horizontal)^2 = (Overall Displacement)^2
Root(Vertical^2 + Horizontal^2) = Magnitude of Displacement
B)
Angle = Tan(Vertical/Horizontal)
To find the magnitude and direction of the resultant displacement, we need to break down the given distances into components and then calculate the vector sum.
a) To find the magnitude of the displacement, we can use the Pythagorean theorem. Let's break down the given distances into their x and y components.
For the first leg (13.3 mi northeast):
The x-component (13.3 mi cos 45°) = 13.3 mi * cos(45°) = 9.4 mi
The y-component (13.3 mi sin 45°) = 13.3 mi * sin(45°) = 9.4 mi
For the second leg (17.3 mi straight south):
The x-component is 0 mi since it is straight south.
The y-component is -17.3 mi since it is in the negative y-direction.
For the third leg (6.85 mi, 30.3° north of west):
The x-component (-6.85 mi cos 30.3°) = -6.85 mi * cos(30.3°) = -5.94 mi
The y-component (6.85 mi sin 30.3°) = 6.85 mi * sin(30.3°) = 3.49 mi
Now, let's add the x and y components separately and then find the magnitude of the resultant displacement using the Pythagorean theorem:
x-component = 9.4 mi + 0 mi + (-5.94 mi) = 3.46 mi (rounded to 2 decimal places)
y-component = 9.4 mi + (-17.3 mi) + 3.49 mi = -4.41 mi (rounded to 2 decimal places)
Magnitude of the resultant displacement = sqrt((3.46 mi)^2 + (-4.41 mi)^2) = sqrt(11.99 mi^2 + 19.44 mi^2) = sqrt(31.43 mi^2) = 5.61 mi (rounded to 2 decimal places)
So, the magnitude of the resultant displacement is approximately 5.61 miles.
b) To find the direction of the resultant displacement, we can use trigonometry. The angle can be found using the inverse tangent function:
Angle = tan^(-1)(y-component / x-component) = tan^(-1)(-4.41 mi / 3.46 mi) = -52.7° (rounded to 1 decimal place)
Since the angle is negative, we can express the direction counterclockwise by subtracting it from 360°:
Direction of resultant displacement = 360° - 52.7° = 307.3° (rounded to 1 decimal place)
So, the direction of the resultant displacement is approximately 307.3° counterclockwise relative to the positive x-axis.