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, and sorry for reposting.
This needs to be done on paper, which we can't do.
I don't get your answers. I used a calculator and got 60.327km at 8.4985 E of N, which is near your answers.
Which means avg velocity is 69.57km/hr at 8.4957 E of N. The angle in N of E would be 90 -8.4957
http://www.1728.org/vectors.htm
adding -75 at 208 to 27.1 at 256= 60.327 at 8.49 E of N
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 displacement.
To correctly solve the problem, let's break it down step by step:
(a) To draw the initial and final position vectors, we need to find the coordinates of the points Cornwall, Atkins Glen, and Illium, and then plot them on a graph.
First, let's determine the coordinates of Cornwall, Atkins Glen, and Illium by breaking down the given information:
- Cornwall is 75 km from Illium in a direction 26° west of south.
This means that Cornwall is 75 km away from Illium, and the angle between the line connecting Cornwall and Illium and the south direction is 26° west.
- Atkins Glen is 27.1 km from Illium in a direction 16° south of west.
This means that Atkins Glen is 27.1 km away from Illium, and the angle between the line connecting Atkins Glen and Illium and the west direction is 16° south.
Next, let's plot these points on a graph. Choose a suitable scale and orientation for the graph, and label the x-axis as east-west and the y-axis as north-south. Place the origin at Illium.
Now draw the vector representing the initial position from Illium to Cornwall. This vector should be 75 units long and have an angle of 26° west of south with respect to the south direction.
Similarly, draw the vector representing the final position from Illium to Atkins Glen. This vector should be 27.1 units long and have an angle of 16° south of west with respect to the west direction.
(b) To find the displacement during the trip, we need to calculate the sum of the initial and final position vectors.
Start by breaking down both vectors into their x and y-components. Remember that the north direction is positive on the y-axis, and the east direction is positive on the x-axis.
For the initial position vector:
- The angle between the vector and the south direction is 26° west, which means the angle with respect to the positive y-axis would be 26° east.
- Calculate the y-component by multiplying the magnitude (75 km) by the sine of the angle (26°).
- Calculate the x-component by multiplying the magnitude by the cosine of the angle.
For the final position vector:
- The angle between the vector and the west direction is 16° south.
- Calculate the y-component by multiplying the magnitude (27.1 km) by the cosine of the angle.
- Calculate the x-component by multiplying the magnitude by the sine of the angle (-16° since it's south of west; negative because it's towards the left).
Now combine the x and y-components of both vectors by adding them together to get the components of the final displacement vector.
Finally, calculate the magnitude of the displacement vector using the Pythagorean theorem: sqrt(dx^2 + dy^2). This will give you the magnitude of the displacement in km.
(c) To find Peggy's average velocity for the trip, use the formula: average velocity = displacement / time.
Now that you have the displacement from part (b) and the time given in the problem (52 min), you can calculate the average velocity.
Calculate the magnitude of the average velocity using the formula: magnitude = displacement / time. This will give you the average velocity in km/h.
To find the direction of Peggy's average velocity, use trigonometry to determine the angle between the positive x-axis (east direction) and the displacement vector. This will give you the direction in degrees north of east.
By following these steps, you should be able to find the correct answers for parts (b) and (c).