While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 260 m east, 195 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.

While exploring a cave, a spelunker starts at the entrance and moves the following distances. She goes 75.0 m north, 250 m east, 116 m at an angle 30.0° north of east, and 156 m south. Find the resultant displacement from the cave entrance. The figure below suggests the situation but is not drawn to scale.

Let x=east, y=north

and 0° towards the +x-axis (due East)
and 90° = due north.
Separate each displacement into the x- and y-components and add.
distance angle x y
75m N=90° 75cos(90°) 75sin(90°)
260m E=0° 260cos(0) 260sin(0)
195m 30°N of E = 30° 195cos(30°) 195sin(30°)
...

Add up algebraically the x- and y-components. The sum will be the x- and y-components of the resultant displacement vector.

To find the resultant displacement from the cave entrance, we need to combine the individual displacements in both the north-south and east-west directions.

Let's break down the given displacements:

1. The spelunker moves 75.0 m north.
2. The spelunker then moves 260 m east.
3. The spelunker then moves 195 m at an angle of 30.0° north of east.
4. Finally, the spelunker moves 150 m south.

Step 1: Convert the displacements into their respective north (+) and south (-) components:

1. The north displacement is 75.0 m.
2. The east displacement is 260 m (no north/south component).
3. The displacement at an angle of 30.0° north of east can be broken down into vertical and horizontal components using trigonometry. The vertical component is given by 195 m * sin(30.0°) and the horizontal component is given by 195 m * cos(30.0°).
4. The south displacement is -150 m (negative since it is in the opposite direction from north).

Step 2: Calculate the combined north and south displacements:

North displacement = 75.0 m

South displacement = -150 m

Total north-south displacement = 75.0 m + (-150 m) = -75.0 m

Step 3: Calculate the combined east and west displacements:

East displacement = 260 m

There are no west displacements, so the total east-west displacement remains 260 m.

Step 4: Calculate the resultant displacement using the Pythagorean theorem:

Resultant displacement = √((east-west displacement)^2 + (north-south displacement)^2)

Resultant displacement = √((260 m)^2 + (-75.0 m)^2)

Resultant displacement = √(67600 m^2 + 5625 m^2)

Resultant displacement = √(73225 m^2)

Resultant displacement ≈ 270.49 m

Therefore, the resultant displacement from the cave entrance is approximately 270.49 meters.

A cave explorer is called a speleologist.