While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 225 m east, 160 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.
You have four vectors.
For each one resolve them into North and east components.
Adding the North components.
N= 75 + 0 + 125 sin(30) - 150 = - 12.5 m
And adding the east components we get
E = 0 + 250 + 125 cos(30) + 0 = 358.25
The magnitude of the result is found by Pythagoras' theory
C = sqrt (a^2 + b^2)
= 358 m to three significant figures.
And the angle is south of east.
found by arctan (12.5/358.25)
= 2 degrees south of East.
To find the resultant displacement from the cave entrance, we need to calculate the horizontal and vertical components of each movement and then add them up.
Let's break down each movement:
1. 75.0 m north: This is a purely vertical movement, so the horizontal component is 0 and the vertical component is +75.0 m.
2. 225 m east: This is a purely horizontal movement, so the horizontal component is +225 m and the vertical component is 0.
3. 160 m at an angle 30° north of east: To find the horizontal and vertical components, we need to use trigonometry. The horizontal component can be found by multiplying the magnitude (160 m) by the cosine of the angle (30°), so the horizontal component is +138.564 m. The vertical component can be found by multiplying the magnitude (160 m) by the sine of the angle (30°), so the vertical component is +80 m.
4. 150 m south: This is a purely vertical movement, so the horizontal component is 0 and the vertical component is -150 m.
Now we can add up the horizontal and vertical components:
Horizontal displacement = 225 m (east) + 138.564 m (east) = +363.564 m (east)
Vertical displacement = 75.0 m (north) + 80 m (north) - 150 m (south) = +5.0 m (north)
Therefore, the resultant displacement from the cave entrance is +363.564 m east and +5.0 m north.