A spelunker is surveying a cave. She follows a passage 150m straight west, then 270m in a direction 45^\circ east of south, and then 280 {\rm m} at 30^\circ east of north. After a fourth unmeasured displacement, she finds herself back where she started.Use a scale drawing to determine the magnitude of the fourth displacement?
After doing all the calculations below, I just realized you just needed a scale drawing.
Oh, well, no harm done
Get a nice sheet of graph paper, scale the units to the following
150 m --> 15/2 = 7.5 cm
270 m --> 27/2 = 13.5 cm
280 m --> 28/2 = 14 cm , to fit on a standard sheet of paper.
You will need a compass and a protractor.
Most of your sketch will be in the third and fourth quadrant of the grid, so put your origin near the top part of the graph paper about 1/2 of the way over.
1. from your origin, draw a horizontal line 7.5 cm long
2. at the end of that line draw a line down at S 45°E.
3. set your compass to 13.5 cm and mark off that line
4. from that end point, draw a line making an angle of 75° with the previous line
5. set your compass to 14 cm and mark off this new line.
6. Join this point to the origin
7. measure that line, it should be appr. 9.2 cm (if my work below does not contain any stupid arithmetic errors).
I think we can solve this using vectors.
any vector (x,y) can be described by (rcosØ, rsinØ), where r is its magnitude and Ø is the angle it forms with the positive x-axis
so we have to convert our different paths
150m straight west --> (150cos180, 150sin180) = (-150, 0)
270m in a direction 45^\circ east of south
= (270cos315°, 270sin315) = (190.919 , -190.919)
280 {\rm m} at 30^\circ east of north
= (280cos60, 280sin60) = (140 , 242.49)
so (-150, 0) + (190.919 , -190.919) + (140 , 242.49) + (x,y) = (0,0)
-150 + 190.919 + 140 + x = 0
x = -180.919
0 -190.919 + 242.49 + y = 0
y = -51.571
(x,y) = (-180.919, -51.571)
magnitude = √(x^2 + y^2) = 188.13 m or 188 m to the nearest metre
To determine the magnitude of the fourth displacement, we need to create a scale drawing of the spelunker's journey and use vector addition.
Let's start by drawing a coordinate system. We can use a straight line for the western direction and label it as 150m.
Next, we need to draw the second displacement, which is 270m at a direction 45 degrees east of south. To do this, we draw a line segment at a 45-degree angle downwards from the end of the first displacement. The length of this line segment represents 270m.
Then, we draw the third displacement, which is 280m at a direction 30 degrees east of north. To do this, we draw a line segment at a 30-degree angle upwards from the end of the second displacement. The length of this line segment represents 280m.
Lastly, we need to draw the fourth displacement, which is unmeasured. Since the spelunker finds herself back where she started, the final displacement must complete a closed triangle. We can draw a line segment from the end of the third displacement to the starting point.
Now, we can use vector addition to find the magnitude of the fourth displacement. Starting from the starting point, we connect the start and end points of all the displacements to form a triangle.
To measure the magnitude of the fourth displacement, we draw a line segment from the end of the third displacement to the start of the first displacement (forming a triangle).
With the scale drawing, we can measure the length of the line segment representing the fourth displacement using a ruler.
Note: Since a scale was not given in the problem, we cannot determine the actual length in meters without additional information. However, we can determine the length relative to the other displacements in the scale drawing.
To determine the magnitude of the fourth displacement, we need to create a scale drawing of the spelunker's movements.
1. Start by drawing a horizontal line to represent the westward movement of 150m. Label this line as "West 150m."
2. Next, draw a second line originating from the end of the first line at an angle of 45 degrees below the south direction. This line should measure 270m. Label it as "45° East of South 270m."
3. From the end of the second line, draw a third line at an angle of 30 degrees above the north direction. This line should measure 280m. Label it as "30° East of North 280m."
4. Finally, draw a fourth line connecting the endpoint of the third line back to the starting point. The value of this line will represent the magnitude of the fourth displacement.
Measure the length of the fourth line on the scale drawing using a ruler, and convert the value to the appropriate unit (e.g., meters).