While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 245 m east, 210 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.
To find the resultant displacement from the cave entrance, we need to add up all the individual displacements. We can represent each displacement as a vector, where the magnitude represents the distance and the direction represents the direction of movement.
Let's break down the given displacements into components.
1) The displacement of 75.0 m north can be represented as:
- Displacement in the x-direction: 0 m (no movement in the x-direction)
- Displacement in the y-direction: 75.0 m north
2) The displacement of 245 m east can be represented as:
- Displacement in the x-direction: 245 m east
- Displacement in the y-direction: 0 m (no movement in the y-direction)
3) The displacement of 210 m at an angle 30.0° north of east can be broken down into its x and y-components using trigonometry:
- Displacement in the x-direction: 210 m * cos(30.0°)
- Displacement in the y-direction: 210 m * sin(30.0°)
4) The displacement of 150 m south can be represented as:
- Displacement in the x-direction: 0 m (no movement in the x-direction)
- Displacement in the y-direction: -150 m south
Now, we can add up the x and y-components of all the displacements to find the resultant displacement:
x-component = (245 m) + (210 m * cos(30.0°)) + (0 m) + (0 m) = 245 m + 181.5 m + 0 m + 0 m = 426.5 m east
y-component = (75.0 m) + (0 m) + (210 m * sin(30.0°)) + (-150 m) = 75 m + 0 m + 105 m - 150 m = 30 m north
The resultant displacement can be found using the Pythagorean theorem:
resultant displacement = sqrt( (x-component)^2 + (y-component)^2 )
= sqrt( (426.5 m)^2 + (30 m)^2 )
≈ 427.7 m
Therefore, the resultant displacement from the cave entrance is approximately 427.7 m, in a direction of approximately 7.0° east of north.