A bird watcher meanders through the woods, walking 1.13 km due east, 1.01 km due south, and 3.95 km in a direction 49.8 ° north of west. The time required for this trip is 1.965 h. Determine the magnitudes of the bird watcher's (a) displacement and (b) average velocity.
a. D = 1.13km[0o] + 1.01km[270o] + 3.95km[130.2o] = (1.13+3.95*Cos130.2) + (1.01*sin270+3.95*sin130.2)i =
-1.42 - 1.009i = 1.74km[35.4o W. of S.].
b. D/t = 1.74km[35.4o W. of S.]/1.966h = 0.885km/h[35.4o W. of S.].
To determine the magnitudes of the bird watcher's displacement and average velocity, we need to use vector addition and some basic formulas.
(a) Displacement:
To find the displacement, we need to find the net vector from the initial position to the final position of the bird watcher.
Step 1: Convert the distances travelled into vectors.
- The first leg, walking 1.13 km due east, can be represented by a vector with a magnitude of 1.13 km pointing in the positive x-direction.
- The second leg, walking 1.01 km due south, can be represented by a vector with a magnitude of 1.01 km pointing in the negative y-direction.
- The third leg, walking 3.95 km at an angle of 49.8° north of west, can be separated into its x and y components using trigonometry.
- The x-component is given by cos(49.8°) * 3.95 km in the negative x-direction.
- The y-component is given by sin(49.8°) * 3.95 km in the positive y-direction.
Step 2: Add the vectors.
To add the vectors, we need to sum their respective x and y components.
- The x-component is the sum of the x-components of each leg.
- The y-component is the sum of the y-components of each leg.
Step 3: Calculate the magnitude of the displacement vector.
The magnitude of the displacement vector is given by the formula:
|displacement| = √(x^2 + y^2)
(b) Average Velocity:
To find the average velocity, we need to divide the displacement vector by the time taken.
Step 4: Calculate the magnitude of the average velocity vector.
The magnitude of the average velocity vector is given by the formula:
|average velocity| = |displacement| / time
Now let's calculate the magnitudes of the bird watcher's displacement and average velocity.
Step 1: Convert the distances travelled into vectors.
- The first leg has a vector representation of (1.13, 0) km.
- The second leg has a vector representation of (0, -1.01) km.
- The third leg can be separated into its x and y components:
- The x-component is -1 * cos(49.8°) * 3.95 km.
- The y-component is sin(49.8°) * 3.95 km.
Step 2: Add the vectors.
- The sum of the x-components is: -1 * cos(49.8°) * 3.95 km.
- The sum of the y-components is: -1.01 km + sin(49.8°) * 3.95 km.
Step 3: Calculate the magnitude of the displacement.
|displacement| = √((-1 * cos(49.8°) * 3.95)^2 + (-1.01 + sin(49.8°) * 3.95)^2)
Step 4: Calculate the magnitude of the average velocity.
|average velocity| = |displacement| / 1.965 h
Now you can substitute the numerical values into the formulas to find the answers.