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 10.0 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 determine the distance and direction between Ricardo and Jane after they walk away in different directions, we can use vector addition.
First, let's break down the displacement of each person into its components.
For Ricardo:
- Distance walked: 28.0 meters
- Direction: 60.0 degrees west of north
For Jane:
- Distance walked: 10.0 meters
- Direction: 30.0 degrees south of west
We can convert the directions into their respective components by using trigonometry.
For Ricardo:
- Vertical component: 28.0 * sin(60.0) = 24.2 meters (north direction)
- Horizontal component: 28.0 * cos(60.0) = 14.0 meters (west direction)
For Jane:
- Vertical component: 10.0 * sin(30.0) = -5.0 meters (south direction)
- Horizontal component: 10.0 * cos(30.0) = -8.7 meters (west direction)
Now, we can find the total displacement by adding the horizontal and vertical components separately.
Horizontal displacement: 14.0 meters - 8.7 meters = 5.3 meters (west direction)
Vertical displacement: 24.2 meters - 5.0 meters = 19.2 meters (north direction)
To find the distance between Ricardo and Jane, we can use the Pythagorean theorem.
Distance = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
Distance = sqrt((5.3 meters)^2 + (19.2 meters)^2)
Distance ≈ sqrt(28.09 + 368.64)
Distance ≈ sqrt(396.73)
Distance ≈ 19.92 meters
Lastly, we can find the direction between Ricardo and Jane by using inverse tangent (arctan) to find the angle.
Direction = arctan(vertical displacement / horizontal displacement)
Direction = arctan(19.2 meters / 5.3 meters)
Direction ≈ arctan(3.63)
Therefore, Ricardo and Jane are approximately 19.92 meters apart in a direction of approximately 75.9 degrees west of north.