A spelunker is surveying a cave. She follows a passage 120 m straight west, then 210 m in a direction 45∘ east of south, and then 280 m at 30∘ east of north. After a fourth unmeasured displacement, she finds herself back where she started.
A. Use a scale drawing to determine the magnitude of the fourth displacement.
B. Determine the direction of the fourth displacement. (....) south of west
Unless otherwise stated, all angles are measured CCW from the
+x-axis.
D = -120 + 210[315o] + 280[60o].
X = -120 + 210*Cos315 + 280*Cos60
= 168.5 m.
Y=210*sin315 + 280*sin60 = 94 m.
Tan A = Y/X = 94/168.5 = 0.55786
A = 29.2o
D = X/Cos A = 168.5/Cos29.2 = 193 m[29o].
b. Direction = 29 + 180 = 209o =
29o S. of W.
A. To determine the magnitude of the fourth displacement, we can use the concept of vector addition.
1. Draw a scale drawing to represent the given displacements. Choose a suitable scale that allows you to draw the cave and the displacements accurately.
2. Start by drawing a line segment to represent the first displacement of 120 m straight west. Label this segment as 120 m.
3. From the end point of the first displacement, draw a line segment 210 m in a direction 45° east of south. Label this segment as 210 m.
4. From the end point of the second displacement, draw a line segment 280 m at 30° east of north. Label this segment as 280 m.
5. Finally, connect the end point of the third displacement to the starting point to complete the triangle.
Now, measure the length of this fourth side of the triangle on your scale drawing. This magnitude represents the magnitude of the fourth displacement.
B. To determine the direction of the fourth displacement, measure the angle between the fourth side and the west direction on your scale drawing. This angle represents the direction of the fourth displacement in relation to the west direction.
So, on your scale drawing, measure the angle between the fourth side and the west direction and express the direction as "X degrees south of west".
To solve this problem, we can use vector addition to find the magnitude and direction of the fourth displacement.
To start, let's draw a scale diagram to represent the spelunker's movement. We'll assume that 1 cm on the diagram represents 10 m in real life.
1. Draw a line segment representing the first displacement: 120 m straight west. This line segment will be 12 cm long and will point directly to the left.
2. From the end point of the first displacement, draw a second line segment. This line segment represents the second displacement: 210 m at a direction 45∘ east of south. Using a protractor, measure an angle of 45∘ from the south direction and draw a line segment 21 cm long in that direction.
3. From the end point of the second displacement, draw a third line segment. This line segment represents the third displacement: 280 m at 30∘ east of north. Measure an angle of 30∘ from the north direction and draw a line segment 28 cm long in that direction.
4. The spelunker finds herself back where she started after these three displacements. At this point, draw the fourth displacement as a line segment from the end point of the third displacement back to the starting point.
Now that we have the scale diagram, we can measure the magnitude and direction of the fourth displacement.
A. To find the magnitude of the fourth displacement, measure the length of the line segment representing it on the scale diagram and convert it back to real-life units. Let's assume the length on the diagram is 10 cm, which represents 100 m. Thus, the magnitude of the fourth displacement is 100 m.
B. To find the direction of the fourth displacement, measure the angle between the line segment representing it and the west direction on the scale diagram. Let's assume the angle is 60∘. The direction of the fourth displacement is then 60∘ south of west.
Therefore, the magnitude of the fourth displacement is 100 m, and its direction is 60∘ south of west.