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.
1. a) What is the magnitude of his resultant displacement?
2. b) What is the direction of his resultant displacement? Express your answer counterclockwise relative to the positive x-axis.
dfgcaer
To solve this problem, we will need to use vector addition to find the resultant displacement of the man's journey.
First, we need to break down each movement into its x and y components. We will consider east as the positive x-direction and north as the positive y-direction.
1. Convert the distances traveled in each direction into their x and y components:
a) The man drives 13.3 mi northeast. The northeast direction can be divided into equal north and east components. Using basic trigonometry, we can find that the northeast movement can be broken down into:
- x-component: 13.3 mi * cos(45°)
- y-component: 13.3 mi * sin(45°)
b) The man then drives 17.3 mi straight south. Since this is a straight south movement, there is no x-component, and the y-component is -17.3 mi.
c) Finally, the man drives 6.85 mi in a direction 30.3° north of west. To find the x and y components, we need to resolve this movement into its north and west components:
- x-component: 6.85 mi * cos(30.3°)
- y-component: 6.85 mi * sin(30.3°)
2. Calculate the total x and y components by adding up the individual x and y components obtained from each movement.
3. Use the Pythagorean theorem to find the magnitude of the resultant displacement:
magnitude = √(x^2 + y^2)
4. Calculate the angle of the resultant displacement relative to the positive x-axis using trigonometry:
angle = atan2(y, x)
Now, let's calculate the answers:
a) Magnitude of the resultant displacement:
magnitude = √[(x-component1 + x-component2 + x-component3)^2 + (y-component1 + y-component2 + y-component3)^2]
Substituting the values, we get:
magnitude = √[(13.3 mi * cos(45°) + 0 mi + 6.85 mi * cos(30.3°))^2 + (13.3 mi * sin(45°) - 17.3 mi + 6.85 mi * sin(30.3°))^2]
Simplifying and calculating the magnitude will give you the answer.
b) Direction of the resultant displacement:
angle = atan2(y-component1 + y-component2 + y-component3, x-component1 + x-component2 + x-component3)
Substituting the values, we get:
angle = atan2((13.3 mi * sin(45°) - 17.3 mi + 6.85 mi * sin(30.3°)), (13.3 mi * cos(45°) + 0 mi + 6.85 mi * cos(30.3°)))
Calculating the angle will give you the direction of the resultant displacement.
Remember to convert the angle to a counterclockwise direction relative to the positive x-axis.