While exploring a cave, a spelunker starts at the entrance and moves the following distances: 75.0 m north, 290 m east, 180 m at an angle 30.0° north of east, and 150 m south. Find the resultant displacement from the cave entrance.
break it down to horizontal (x) and vertical (y) components!
horizontal
x = 290m + 180(cos30)m
vertical
y = 75m + 180(sin30)m -180 m
perform Pythagorean theorem
sqrt(x^2 + y^2)=?
displacement also has direction:
tan^-1(y/x)=?
To find the resultant displacement from the cave entrance, we need to calculate the net displacement taking into account both the distances and angles involved.
First, let's break down the given distances and angles:
1. 75.0 m north: This is a displacement in the positive y-direction (upwards on a coordinate plane).
2. 290 m east: This is a displacement in the positive x-direction (to the right on a coordinate plane).
3. 180 m at an angle 30.0° north of east: We need to convert this displacement into its x and y components. The angle provided tells us that the displacement is directed north of east. To find the components, we use trigonometry. The x-component can be calculated by multiplying the magnitude (180 m) by the cosine of the angle (30°), and the y-component can be found by multiplying the magnitude by the sine of the angle.
x-component = 180 * cos(30°)
y-component = 180 * sin(30°)
4. 150 m south: This is a displacement in the negative y-direction (downwards on a coordinate plane).
Now, let's calculate the x and y components of the displacement:
x-component = 290 m + 180 m * cos(30°)
y-component = 75.0 m - 180 m * sin(30°) - 150 m
Finally, we can calculate the magnitude and direction of the resultant displacement using these components:
magnitude = sqrt((x-component)^2 + (y-component)^2)
direction = arctan(y-component / x-component)
By plugging in the values and performing the calculations, you can find the resultant displacement from the cave entrance.