Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 28.0 in a direction 60.0 west of north. Jane walks 11.0 in a direction 30.0 south of west. They then stop and turn to face each other
What is the distance between them?
In what direction should Ricardo walk to go directly toward Jane?
To find the direction they are facing, we can use the concept of vectors. A vector has both magnitude (length) and direction. In this case, Ricardo's displacement vector is 28.0 at an angle of 60.0° west of north, and Jane's displacement vector is 11.0 at an angle of 30.0° south of west.
To determine their combined displacement vector, we can break down each vector into its horizontal (x) and vertical (y) components.
For Ricardo's vector:
Magnitude: 28.0
Angle: 60.0° west of north
Using trigonometry, we can find the x and y components of Ricardo's vector:
x-component = magnitude * cos(angle) = 28.0 * cos(60.0°) ≈ 14.0
y-component = magnitude * sin(angle) = 28.0 * sin(60.0°) ≈ 24.2
Similarly, for Jane's vector:
Magnitude: 11.0
Angle: 30.0° south of west
Again, using trigonometry:
x-component = magnitude * cos(angle) = 11.0 * cos(30.0°) ≈ 9.5
y-component = magnitude * sin(angle) = 11.0 * sin(30.0°) ≈ 5.5
Now, to find their combined displacement, we add the x-components and y-components separately:
Combined x-component = 14.0 + 9.5 ≈ 23.5
Combined y-component = 24.2 + 5.5 ≈ 29.7
To find the magnitude and direction of their combined displacement, we can use the Pythagorean theorem and inverse trigonometric functions. The magnitude can be calculated as:
Magnitude = √(Combined x-component^2 + Combined y-component^2)
= √(23.5^2 + 29.7^2)
≈ 37.7
To find the direction, we can use the inverse tangent function:
Angle = arctan (Combined y-component / Combined x-component)
= arctan (29.7 / 23.5)
≈ 51.9°
Therefore, Ricardo and Jane are facing each other with a combined displacement vector of approximately 37.7 in a direction of 51.9° east of north.