A bird watcher meanders through the woods, walking 0.35 km due east, 0.65 km due south, and 2.05 km in a direction 31.0° north of west. The time required for this trip is 3.50 h. Determine the magnitude and direction (relative to due west) of the bird watcher's. Find the displacement and km(magnitude) and find the average velocity in km/h
Break up each leg into N and W components. THe first two are given in those components, so you only have to break up the last. Add all vectors, you get a total N and a total W component.
Avg velocity=totaldisplcement/time.
To find the magnitude and direction of the bird watcher's displacement relative to due west, we can break down the given distances into components.
Let's consider the east direction as positive and the west direction as negative.
Distance walked due east = 0.35 km in the positive x-direction.
Distance walked due south = 0.65 km in the negative y-direction.
To find the distance walked in the direction 31.0° north of west, we need to break it down into x and y components.
Distance walked at 31.0° north of west = 2.05 km
x-component = 2.05 km * cos(31.0°)
y-component = 2.05 km * sin(31.0°)
x-component = 2.05 km * cos(31.0°) = 1.753 km (rounded to three decimal places)
y-component = 2.05 km * sin(31.0°) = 1.038 km (rounded to three decimal places)
To find the total displacement in the x and y directions, we sum up the x and y components.
Total displacement in the x-direction = 0.35 km + 1.753 km = 2.103 km
Total displacement in the y-direction = -0.65 km + 1.038 km = 0.388 km
To find the magnitude of displacement (d) using Pythagoras' theorem:
d = √(dx² + dy²)
d = √(2.103 km)² + (0.388 km)²
d = √(4.421209) + (0.150544)
d = √(4.571753)
d = 2.14 km (rounded to two decimal places)
To find the direction (ϴ) relative to due west, we can use the arctan function:
ϴ = atan(dy/dx)
ϴ = atan(0.388 km/2.103 km)
ϴ = atan(0.184351)
ϴ = 10.3° (rounded to one decimal place)
Therefore, the magnitude of the bird watcher's displacement is 2.14 km, and the direction relative to due west is 10.3°.
To find the average velocity, we divide the total displacement by the time taken:
Average velocity = Total displacement / Time taken
Total displacement = √((2.103 km)² + (0.388 km)²) = 2.14 km (rounded to two decimal places)
Time taken = 3.50 hours
Average velocity = 2.14 km / 3.50 hours
Average velocity = 0.611 km/h (rounded to three decimal places)
Therefore, the bird watcher's average velocity is approximately 0.611 km/h, considering both magnitude and direction.
To find the magnitude and direction (relative to due west) of the bird watcher's final displacement, we can break down their movements into horizontal (east/west) and vertical (north/south) components.
1. Horizontal Component:
The bird watcher walks 0.35 km due east and then 2.05 km at an angle 31.0° north of west. To find the total horizontal displacement, we need to calculate the vector sum of these two distances.
Horizontal displacement = 0.35 km + 2.05 km * cos(31.0°)
2. Vertical Component:
The bird watcher walks 0.65 km due south. This contributes to the vertical displacement.
Vertical displacement = -0.65 km
3. Magnitude of Displacement:
The magnitude of displacement is the distance between the starting and ending points. We can use the Pythagorean theorem to find the magnitude:
Magnitude = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
4. Direction (Relative to Due West):
To find the direction of displacement relative to due west, we can use trigonometric functions. The angle can be determined using:
Angle = atan2(vertical displacement, horizontal displacement)
5. Average Velocity:
Average velocity is the total displacement divided by the time taken. We can find the average velocity using:
Average velocity = Displacement / Time
Now, let's calculate the values:
Horizontal displacement = 0.35 km + 2.05 km * cos(31.0°) (using a calculator)
Vertical displacement = -0.65 km
Magnitude = sqrt((Horizontal displacement)^2 + (Vertical displacement)^2) (using a calculator)
Direction (Relative to Due West) = atan2(Vertical displacement, Horizontal displacement) (using a calculator)
Average velocity = Magnitude / Time
Substituting the given values:
Time = 3.50 h
Solving for the above equations will give us the final results.