A bird watcher meanders through the woods, walking 0.544 km due east, 0.925 km due south, and 3.58 km in a direction 17.3 ° north of west. The time required for this trip is 1.957 h. Determine the magnitudes of the bird watcher's (a) displacement and (b) average velocity.
To determine the magnitudes of the bird watcher's displacement and average velocity, we can use vector addition and the given information.
(a) Displacement:
To find the displacement, we need to find the net displacement vector. We can find it by adding all the displacement vectors together.
First, let's convert the distances into Cartesian coordinates using their respective directions:
0.544 km due east = +0.544 km in the x-axis (eastward direction)
0.925 km due south = -0.925 km in the y-axis (southward direction)
3.58 km in a direction 17.3° north of west = 3.58 km * cos(17.3°) in the x-axis and 3.58 km * sin(17.3°) in the y-axis.
Calculating the Cartesian coordinates:
x-component = 3.58 km * cos(17.3°) = +3.43 km (to the right)
y-component = -3.58 km * sin(17.3°) = -0.976 km (to the south)
Now we can find the net displacement vector by adding the x and y components:
Net displacement vector = x-component + y-component = +3.43 km - 0.976 km.
The magnitude of the displacement is the straight-line distance from the initial position to the final position, which can be found using the Pythagorean theorem:
Magnitude (displacement) = sqrt( x^2 + y^2 ) where x and y are the components of the net displacement vector.
Plugging in the values, we get:
Magnitude (displacement) = sqrt( (3.43 km)^2 + (-0.976 km)^2 )
(b) Average Velocity:
Average velocity is defined as the total displacement divided by the total time taken.
The magnitude of the average velocity is the ratio of the magnitude of displacement to the total time:
Magnitude (average velocity) = Magnitude (displacement) / (time taken)
Given that the time taken is 1.957 hours, the magnitude of the average velocity is:
Magnitude (average velocity) = Magnitude (displacement) / 1.957 hours
Now we can substitute the value of the magnitude of displacement into the equation to find the magnitude of the average velocity.
By computing these values, we can determine the magnitudes of the bird watcher's displacement and average velocity.
0.544 km E = <0.544,0.000>
0.925 km S = <0.000,-0.925>
3.58 km W17.3°N = <-3.418,1.065>
Adding them all up, the final location is <-2.874,0.140>
Now you should be able to figure the final displacement and thus the average velocity.