While exploring a cave, a spelunker starts at the entrance and moves the following distances. She goes 75.0 m north, 250 m east, 109 m at an angle θ = 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.
magnitude :
direction : ° south of east
treating the vectors as complex numbers, we have
75i + 250 + 109cis(pi/6) - 156i
= 344.4 - 26.5i
or, in polar form,
345.4 @ E4.4°S
i don't understand the direction
To find the resultant displacement, we need to determine the magnitude and direction.
Step 1: Determine the north and east components of the displacement.
Given information:
- 75.0 m north
- 250 m east
- 109 m at an angle θ = 30.0° north of east
- 156 m south
To find the north component, we need to consider the displacements that are northward:
- 75.0 m north
- 109 m at an angle θ = 30.0° north of east
Using trigonometry, we can determine the north component of the angled displacement:
north component = 109 m * sin(30°) = 54.5 m
To find the east component, we need to consider the displacements that are eastward:
- 250 m east
- 109 m at an angle θ = 30.0° north of east
Using trigonometry, we can determine the east component of the angled displacement:
east component = 109 m * cos(30°) = 94.6 m
Now, add up the north and south components:
north component = 75.0 m - 156 m = -81.0 m
Note: The negative sign indicates a southward direction.
Step 2: Calculate the resultant displacement magnitude.
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem:
resultant magnitude = √(east component^2 + north component^2)
resultant magnitude = √(94.6 m^2 + (-81.0 m)^2) = √(8943.16 m^2 + 6561 m^2) = √15504.16 m^2 = 124.5 m
Step 3: Calculate the resultant displacement direction.
To find the direction of the resultant displacement, we can use trigonometric functions:
resultant direction = atan(north component / east component)
resultant direction = atan(-81.0 m / 94.6 m) = atan(-0.857) ≈ -41.0°
Note: The negative sign indicates a direction south of east.
Therefore, the resultant displacement is:
Magnitude: 124.5 m
Direction: 41.0° south of east