A bird watcher meanders through the woods, walking 1.10 km due east, 0.624 km due south, and 1.09 km in a direction 18.2 ° north of west. The time required for this trip is 1.046 h. Determine the magnitudes of the bird watcher's (a) displacement and (b) average velocity.
a. D=1.10km - 0.624i km + 1.09km[161.8o]
X = 1.10 + 1.09*cos161.8 = 0.0645 km. =
64.5 m.
Y = - 0.645 + 1.09*sin161.8 = -0.305 km.
= -305 m.
Tan A = Y/X = -305/64.5 = -4.72868
A = -78.06o CW = 78.06o S. of E.
D = Y/Sin(-78.06) = -305/Sin(-78.06) =
312 m.[-78.06].
b. V = D/t = 312m[-78.06]/1.046h = 298
m/h = 0.298km/h[-78.06o]
Correction:
a. 1.10km[0o] + 0.624km[270o] + 1.09[161.8o]CCW.
X = 1.10 + 0.624*Cos270 + 1.09*Cos161.8 = 771.10 + 0 - 1.035 = 0.0645km = 64.5 m.
Y = 0.624*sin270 + 1.09*sin161.8 =
-0.624 + 0.340 = -0.284km = -284 m.
Tan A = Y/X = 284/64.5 = -4.40310.
A = -77.2o = 77.2o S. of E. = 282.8oCCW
Displacement = Y/sinA = -284/sin282.8 =
291.2 m[282.8o]CCW.
b. V=(d1+d2+d3)/T=(1.10+.624+1.09)/1.046
= 2.69 km/h.
To find the bird watcher's displacement, we need to find the total distance covered in a straight line from the starting point to the ending point.
(a) Displacement:
We can use the Pythagorean theorem to find the displacement.
First, let's find the components of the displacement in the x and y directions:
- In the x-direction: 1.10 km due east - 1.09 km in a direction 18.2° north of west.
To find the x-component of the displacement, we calculate: D_x = 1.10 km - 1.09 km * cos(18.2°)
- In the y-direction: 0.624 km due south + 1.09 km sin(18.2°) in the opposite direction.
To find the y-component of the displacement, we calculate: D_y = -0.624 km + 1.09 km * sin(18.2°)
Now, we can find the magnitude of the displacement using the Pythagorean theorem:
Displacement = √(D_x^2 + D_y^2)
Calculate the values:
D_x = 1.10 km - 1.09 km * cos(18.2°)
D_x = 0.011 km
D_y = -0.624 km + 1.09 km * sin(18.2°)
D_y = 0.197 km
Displacement = √(0.011 km^2 + 0.197 km^2)
Displacement = √(0.000121 km^2 + 0.038809 km^2)
Displacement = √(0.038930 km^2)
Displacement = 0.197 km (rounded to three decimal places)
Therefore, the magnitude of the bird watcher's displacement is 0.197 km.
(b) Average Velocity:
Average velocity is defined as the displacement divided by the total time taken.
Average velocity = Displacement / Time
Average velocity = 0.197 km / 1.046 h
Average velocity = 0.188 km/h (rounded to three decimal places)
Therefore, the bird watcher's average velocity is 0.188 km/h.
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 vector sum of the individual displacements in the x and y directions.
Step 1: Resolve the vectors into their x and y components.
The first vector (1.10 km due east) only has an x-component, which is +1.10 km.
The second vector (0.624 km due south) only has a y-component, which is -0.624 km.
The third vector (1.09 km in a direction 18.2° north of west) needs to be resolved into x and y components. The x-component is -1.09 km * cos(18.2°), and the y-component is +1.09 km * sin(18.2°).
Step 2: Add the x and y components separately.
The x-component is 1.10 km + (-1.09 km * cos(18.2°)).
The y-component is -0.624 km + (1.09 km * sin(18.2°)).
Step 3: Calculate the resultant displacement.
The displacement is the vector sum of the x and y components. To find the magnitude, we use the Pythagorean theorem:
Displacement = sqrt((x-component)^2 + (y-component)^2).
(b) Average Velocity:
Velocity is defined as the displacement divided by the time.
Step 1: Calculate the displacement.
Use the same steps as in part (a).
Step 2: Calculate the average velocity.
Average Velocity = Displacement / Time.
Now, let's calculate:
(a) Displacement:
Step 1: Resolve the vectors into their x and y components.
x-component of the third vector = -1.09 km * cos(18.2°) = -1.041 km.
y-component of the third vector = 1.09 km * sin(18.2°) = 0.323 km.
Step 2: Add the x and y components separately.
x-component = 1.10 km + (-1.041 km) = -0.581 km.
y-component = -0.624 km + (0.323 km) = -0.301 km.
Step 3: Calculate the resultant displacement.
Displacement = sqrt((-0.581 km)^2 + (-0.301 km)^2) = 0.648 km.
Therefore, the magnitude of the bird watcher's displacement is 0.648 km.
(b) Average Velocity:
Step 1: Calculate the displacement.
Using the results from part (a), the magnitude of the displacement is 0.648 km.
Step 2: Calculate the average velocity.
Average Velocity = Displacement / Time = 0.648 km / 1.046 h = 0.619 km/h.
Therefore, the magnitude of the bird watcher's average velocity is 0.619 km/h.