while exploring a cave, a spelunker starts at the entrance and moves the following distances in a horizontal plane. she goes 75.0 m north, 250 m east, 125 m at an angle theta equal to 30.0 north of east, and 150 m south. find her resultant displacement from the cave entrance.
the displacement for each leg of the journey is
(0,75)
(250,0)
(108.25,62.5) since x=125cos30°, y=125sin30°
(0,-150)
add 'em up to get
(358.25,-12.5)
358.5m final displacement.
To find the resultant displacement from the cave entrance, we need to find the horizontal and vertical components of each movement and then sum them up.
First, let's break down each movement into horizontal and vertical components:
Movement 1 (75.0 m north):
Horizontal component = 0 m
Vertical component = +75.0 m (north)
Movement 2 (250 m east):
Horizontal component = +250 m (east)
Vertical component = 0 m
Movement 3 (125 m at an angle θ = 30.0° north of east):
To find the horizontal and vertical components, we can use trigonometry.
Horizontal component = 125 m * cos(30.0°) = +108.253 m (east)
Vertical component = 125 m * sin(30.0°) = +62.50 m (north)
Movement 4 (150 m south):
Horizontal component = 0 m
Vertical component = -150 m (south)
Now, let's sum up the horizontal and vertical components:
Horizontal component: +250 m (east) + 108.253 m (east) + 0 m + 0 m = +358.253 m (east)
Vertical component: +75.0 m (north) + 0 m + 62.50 m (north) - 150 m (south) = -12.5 m (north)
Finally, we can find the resultant displacement by calculating the magnitude (R) and the direction (θ) using the Pythagorean theorem and inverse tangent.
Magnitude (R) = √((Horizontal component)^2 + (Vertical component)^2) = √((358.253)^2 + (-12.5)^2) = 359.29 m
Direction (θ) = arctan(Vertical component / Horizontal component) = arctan(-12.5 m / 358.253 m) = -1.98° (south of east)
Therefore, the spelunker's resultant displacement from the cave entrance is 359.29 m, and it is located 1.98° south of east.
To find the resultant displacement from the cave entrance, we need to calculate the net north-south and east-west distances traveled by the spelunker.
First, let's break down the distances into their respective components:
1. The spelunker goes 75.0 m north.
- This contributes +75.0 m to the net north-south distance.
2. The spelunker goes 250 m east.
- This contributes +250 m to the net east-west distance.
3. The spelunker goes 125 m at an angle θ = 30.0° north of east.
- To find the north and east components, we can use trigonometry.
- The north component is given by: 125 m * cos(30.0°).
- This contributes +108.25 m to the net north-south distance.
- The east component is given by: 125 m * sin(30.0°).
- This contributes +62.50 m to the net east-west distance.
4. The spelunker goes 150 m south.
- This contributes -150 m to the net north-south distance.
Now, let's add up the north-south and east-west distances:
Net north-south distance = 75.0 m + 108.25 m - 150 m = 33.25 m (south)
Net east-west distance = 250 m + 62.50 m = 312.50 m (east)
Finally, we can determine the resultant displacement using the Pythagorean theorem:
Resultant displacement = √[(Net north-south distance)² + (Net east-west distance)²]
Resultant displacement = √[(33.25 m)² + (312.50 m)²]
Resultant displacement ≈ 317.17 m
Therefore, the spelunker's resultant displacement from the cave entrance is approximately 317.17 meters, in the direction of 33.25° south of east.