Peggy drives from Cornwall to Atkins Glen in 52 min. Cornwall is 75 km from Illium in a direction 26° west of south. Atkins Glen is 27.1 km from Illium in a direction 16° south of west. Use Illium as your origin.
(a) Draw the initial and final position vectors. (Do this on paper. Your instructor may ask you to turn in this work.)
(b) Find the displacement during the trip.
magnitude km
direction ° north of east
(c) Find Peggy's average velocity for the trip.
magnitude km/h
direction ° north of east
Could someone help me solve this please I tried but i got it wrong...
I got 64.82 km and 14.57 degrees for part b and 74.79 km and 14.57 degree for part c, but they where wrong; I don't understand why?
Thank you.
We have triangle with two known sides, a =27.1 km and b =75 km, and the angle α between them
α = 90 º - 16 º -26 º = 48º.
Let’s use the law of cosines to solve for the third side (magnitude of displacement)
r = sqrt(a² +b²-2•a•b•cosα) = sqrt(75² +27.1² - 2•75•27.1•cos48º) = 60.34 km.
Then use the law of sines to find the angles opposite to side a = 27.1 km.
a/sinθ = r/sinα .
sin θ = sinα •a/r =sin48•27.1/60.34 = 0.333.
θ = 19.6º,
φ = 90 º – (26 º -19.5 º) = 90 º – 6.5 º = 83.5 º (N of E)
v(average) = r/t = 60.34•60/52 = 69.62 km/hr
Its direction will be the same as the direction of the displacement.
To solve this problem, let's break it down step by step.
(a) Draw the initial and final position vectors:
To represent the initial and final positions, draw a coordinate system with Illium as the origin. Then, draw a vector from the origin to Cornwall and another vector from the origin to Atkins Glen. Label the vectors as "Cornwall" and "Atkins Glen" respectively.
(b) Find the displacement during the trip:
The displacement is a vector that points from the initial position to the final position. To find the displacement vector, we need to subtract the initial position vector from the final position vector.
First, we need to convert the given distances and angles to components in the x and y directions. Let's assume the positive x-axis points east, and the positive y-axis points north.
For Cornwall:
Distance = 75 km
Direction = 26° west of south
To find the x and y components of the Cornwall vector:
x-component = Distance * cos(Direction)
= 75 km * cos(26°)
≈ 67.62 km (rounded to two decimal places)
y-component = Distance * sin(Direction)
= 75 km * sin(26°)
≈ -32.22 km (rounded to two decimal places)
For Atkins Glen:
Distance = 27.1 km
Direction = 16° south of west
To find the x and y components of the Atkins Glen vector:
x-component = Distance * cos(Direction)
= 27.1 km * cos(196°) (since it is south of west)
≈ -26.89 km (rounded to two decimal places)
y-component = Distance * sin(Direction)
= 27.1 km * sin(196°)
≈ -7.48 km (rounded to two decimal places)
Now, let's find the displacement vector by subtracting the initial position vector (Cornwall) from the final position vector (Atkins Glen):
Displacement x-component = (Atkins Glen x-component) - (Cornwall x-component)
= -26.89 km - 67.62 km
≈ -94.51 km (rounded to two decimal places)
Displacement y-component = (Atkins Glen y-component) - (Cornwall y-component)
= -7.48 km - (-32.22 km)
≈ 24.74 km (rounded to two decimal places)
The magnitude of displacement = √(displacement x-component^2 + displacement y-component^2)
= √((-94.51 km)^2 + (24.74 km)^2)
≈ 98.20 km (rounded to two decimal places)
The direction of displacement can be found using trigonometry:
Displacement direction = atan(displacement y-component / displacement x-component)
= atan(24.74 km / -94.51 km)
≈ -14.57° (rounded to two decimal places)
Hence, the displacement during the trip is approximately 98.20 km in a direction 14.57° south of west.
(c) Find Peggy's average velocity for the trip:
Velocity is a vector that represents the change in position over time. The average velocity is calculated by dividing the displacement by the time taken.
Average velocity magnitude = (Displacement magnitude) / (Time taken)
= 98.20 km / (52 min / 60 min/h)
≈ 113.08 km/h (rounded to two decimal places)
The direction of velocity is the same as the direction of displacement, which is 14.57° south of west.
Therefore, Peggy's average velocity for the trip is approximately 113.08 km/h in a direction 14.57° south of west.
If your answers were different, please check whether you made any calculation errors or rounded incorrectly. Double-check the trigonometric functions and make sure to use the correct signs for the components.