A bird watcher meanders through the woods, walking 0.45 km due east, 0.60 km due south, and 2.70 km in a direction 37.0° north of west. The time required for this trip is 4.00 h.
(a) Determine the magnitude and direction (relative to due west) of the bird watcher's displacement. Use kilometers and hours for distance and time, respectively.
(b) Determine the magnitude and direction (relative to due west) of the bird watcher's average velocity.
To solve this problem, we can divide the bird watcher's trip into three separate displacements:
1. The displacement in the east direction: 0.45 km due east
2. The displacement in the south direction: 0.60 km due south
3. The displacement in the northwest direction: 2.70 km at a direction 37.0° north of west
Let's calculate each displacement step-by-step:
1. Displacement in the east direction: Δx1 = 0.45 km due east
2. Displacement in the south direction: Δx2 = -0.60 km due south (negative because we are moving south)
3. Displacement in the northwest direction:
To find the x-component of this displacement, we use the cosine function:
Δx3 = 2.70 km * cos(37.0°)
Δx3 = 2.70 km * 0.7986
Δx3 ≈ 2.16 km
To find the y-component of this displacement, we use the sine function:
Δy3 = 2.70 km * sin(37.0°)
Δy3 = 2.70 km * 0.6018
Δy3 ≈ 1.63 km
4. Total displacement:
To find the magnitude and the direction of the total displacement, we need to sum up all three displacements:
Δx_total = Δx1 + Δx2 + Δx3
Δy_total = Δy3
Δx_total = 0.45 km + (-0.60 km) + 2.16 km
Δx_total = 2.01 km
Δy_total = 1.63 km
To find the magnitude, we use the Pythagorean theorem:
|Δd| = sqrt(Δx_total^2 + Δy_total^2)
|Δd| = sqrt((2.01 km)^2 + (1.63 km)^2)
|Δd| ≈ 2.57 km
The magnitude of the bird watcher's displacement is approximately 2.57 km.
To find the direction relative to due west, we use trigonometry:
θ = arctan(Δy_total / Δx_total)
θ = arctan(1.63 km / 2.01 km)
θ ≈ 39.3°
The direction of the bird watcher's displacement relative to due west is approximately 39.3°.
---
To determine the average velocity, we need to calculate the total displacement and divide it by the total time taken:
Total displacement: Δd = 2.57 km (from previous calculations)
Total time: Δt = 4.00 h
Average velocity: v_avg = Δd / Δt
v_avg = 2.57 km / 4.00 h
v_avg ≈ 0.643 km/h
The magnitude of the bird watcher's average velocity is approximately 0.643 km/h.
The direction of the bird watcher's average velocity relative to due west is the same as the direction of the displacement, which is approximately 39.3°.
To solve part (a) of the problem, we need to find the magnitude and direction of the bird watcher's displacement.
First, let's break down the bird watcher's movements into components.
The 0.45 km due east movement doesn't have any north or west components.
The 0.60 km due south movement doesn't have any east or west components.
The 2.70 km in a direction 37.0° north of west can be broken down into north and west components.
To find the east/west and north/south components, we'll use trigonometry.
The east/west component for the 2.70 km movement is:
East = 2.70 km * cos(37.0°)
The north/south component for the 2.70 km movement is:
North = 2.70 km * sin(37.0°)
Now, let's calculate the total east/west and north/south components:
Total East = East (from 2.70 km movement) + East (from 0.45 km movement)
Total North = North (from 2.70 km movement) - South (from 0.60 km movement)
Lastly, let's find the magnitude and direction of the bird watcher's displacement using the Pythagorean theorem and trigonometry:
Magnitude = √(Total East² + Total North²)
Direction = arctan(Total North/Total East) - 90° (to get the angle relative to due west)
To solve part (b) of the problem, we need to find the magnitude and direction of the bird watcher's average velocity.
The average velocity is defined as the total displacement divided by the time taken:
Average Velocity = Displacement / Time
Magnitude of Average Velocity = Magnitude of Displacement / Time
The direction of the average velocity will be the same as the direction of the displacement.
Now, let's plug in the given values and calculate the answers:
Magnitude of Displacement = √((Total East)² + (Total North)²)
Direction of Displacement = arctan(Total North/Total East) - 90°
Magnitude of Average Velocity = Magnitude of Displacement / Time
Direction of Average Velocity = Direction of Displacement
n = North component = -.6 + 2.7 sin 37
w = West component = -.45 + 2.7 cos 37
magnitude^2 = n^2 + w^2
direction north of west = tan^-1 (n/w)
ns = north speed = n/4
ws = west speed = w/4
now do the same thing for magnitude of velocity
the angle for velocity is the same as for displacement (top and bottom both divided by 4)