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 24.0 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.
1.What is the distance between them?
2.In what direction should Ricardo walk to go directly toward Jane?
To find the distance between Ricardo and Jane, we can use the concept of vector addition. We will calculate the vector displacement for each person and then find the magnitude of the resultant vector.
1. Ricardo's displacement vector can be represented as 24.0 at 60.0° west of north.
To find the horizontal and vertical components of this vector, we use trigonometry.
Horizontal component = 24.0 * cos(60.0°) = 12.0
Vertical component = 24.0 * sin(60.0°) = 20.8
2. Jane's displacement vector can be represented as 10.0 at 30.0° south of west.
To find the horizontal and vertical components of this vector, we use trigonometry.
Horizontal component = 10.0 * cos(30.0°) = 8.7
Vertical component = -10.0 * sin(30.0°) = -5.0
Now let's find the resultant vector by adding the horizontal and vertical components together.
Horizontal component of resultant vector = 12.0 + 8.7 = 20.7
Vertical component of resultant vector = 20.8 - 5.0 = 15.8
To find the magnitude (distance) of the resultant vector:
Magnitude = sqrt((Horizontal component)^2 + (Vertical component)^2)
= sqrt((20.7)^2 + (15.8)^2)
≈ 26.7
Therefore, the distance between Ricardo and Jane is approximately 26.7 units.
Now let's move on to the second question.
2. To find the direction in which Ricardo should walk to go directly toward Jane, we can find the angle between the resultant vector and the horizontal axis.
Angle = atan((Vertical component of resultant vector) / (Horizontal component of resultant vector))
= atan(15.8 / 20.7)
≈ 38.2°
However, since Ricardo is initially facing west of north and wants to go directly toward Jane, he needs to turn 180° in the opposite direction. So the actual direction Ricardo should walk is:
180° + 38.2° = 218.2° east of south
Therefore, Ricardo should walk in a direction approximately 218.2° east of south to go directly toward Jane.