A cave explorer is surveying the cave. He follows a passage 100m straight east, then 50m in the direction 30degree west of north, then 150m at 45degree west of south. After fourth unmeasured displacement he finds himself back where he started. Whats the fourth displacement? In magnitude and direction
yes
To find the fourth displacement, let's break down the given displacements step-by-step and determine the net displacement after the three measured displacements.
Step 1: Passage straight east for 100m.
This displacement is 100m east.
Step 2: Move in the direction 30 degrees west of north for 50m.
To determine the vertical displacement, we need to find the component of this displacement in the north direction. Using trigonometry:
Vertical component = 50m * sin(30 degrees) = 25m.
To find the horizontal component, we need to find the component of this displacement in the west direction:
Horizontal component = 50m * cos(30 degrees) = 43.3m (approx).
The net displacement at this point is 100m east + 43.3m west + 25m north = 100m east - 43.3m west + 25m north.
Step 3: Move 150m at 45 degrees west of south.
To determine the vertical displacement, we need to find the component of this displacement in the south direction. Using trigonometry:
Vertical component = 150m * cos(45 degrees) = 106.1m (approx).
To find the horizontal component, we need to find the component of this displacement in the west direction:
Horizontal component = 150m * sin(45 degrees) = 106.1m (approx).
The net displacement at this point is 100m east - 43.3m west + 25m north - 106.1m south + 106.1m west - 106.1m south = 100m east - 106.1m south.
Since the explorer finds himself back where he started, the net displacement after the three measured displacements is zero. Therefore, the fourth displacement must bring the explorer back to the starting point.
To cancel out the previous net displacement of 100m east - 106.1m south, the fourth displacement should be 100m west + 106.1m north.
Therefore, the fourth displacement is approximately 100m west and 106.1m north.
To find the fourth displacement, let's break down the given displacements step by step:
1. The first displacement is 100m straight east. This means the explorer is 100m to the east from his starting point.
2. The second displacement is 50m in the direction 30 degrees west of north. To determine this displacement, we can visualize a triangle with a vertical side (north) and a slanted side (west of north). The angle between the vertical side and the slanted side is 30 degrees. Using trigonometry, we can find the northward and westward components of this displacement:
Northward component = 50m * sin(30 degrees) = 25m
Westward component = 50m * cos(30 degrees) = 43.3m
So, the explorer is 25m north and 43.3m west from his previous position.
3. The third displacement is 150m at 45 degrees west of south. Again, we can visualize a triangle with a vertical side (south) and a slanted side (west of south). The angle between the vertical side and the slanted side is 45 degrees. Using trigonometry, we can find the southward and westward components of this displacement:
Southward component = 150m * sin(45 degrees) = 106.1m
Westward component = 150m * cos(45 degrees) = 106.1m
So, the explorer is 106.1m south and 106.1m west from his previous position.
From the given information, we know that the explorer ends up back where he started, which means the net displacement is (0m, 0m).
Now, let's calculate the inverse to determine the fourth displacement:
1. Start by summing up the northward and southward components separately:
Northward component: 25m (from the second displacement)
Southward component: -106.1m (opposite direction from the third displacement)
Net vertical displacement: 25m - 106.1m = -81.1m (negative value indicates southward direction)
2. Next, calculate the eastward and westward components separately:
Eastward component: 100m (from the first displacement)
Westward component: 43.3m (from the second displacement) - 106.1m (from the third displacement)
Net horizontal displacement: 100m + 43.3m - 106.1m = 37.2m
Therefore, the fourth displacement is 81.1m south and 37.2m east. In terms of magnitude and direction, the fourth displacement is approximately 89.7m at an angle of 26.2 degrees east of south.
Just add up the displacements and see where you end up. The final displacement is just minus that location.
For example,
50m in the direction 30degree west of north is
<-50sin30°,50cos30°> = <-25.0,43.3>