A spelunker is surveying a cave. She follows a passage 190 m straight west, then 220 m in a direction (base on SET) east of south, and then 280 m at east of north. After a fourth unmeasured displacement, she finds herself back where she started. What is the unmeasured displacement and its direction?
To solve this problem, we need to break down each of the measured displacements into their respective x and y components, and then add them up to find the overall x and y components. Finally, we can use these components to find the magnitude and direction of the unmeasured displacement.
Let's start by breaking down each measured displacement:
1) Displacement 1: 190 m straight west
This displacement is purely in the negative x-direction, so its x-component is -190 m and its y-component is 0 m.
2) Displacement 2: 220 m in a direction (based on SET) east of south
To find the x and y components, we need to determine the angle between the displacement and the positive x-axis. Since it is east of south, the angle would be 180 degrees minus the angle between the displacement and the negative x-axis.
Let's say the angle between the displacement and the negative x-axis is θ. Therefore, the angle between the displacement and the positive x-axis would be 180 - θ.
Now we can find the x and y components using trigonometry.
The x-component is 220 * cos(180 - θ) = -220 * cos(θ) since cos(180 - θ) = -cos(θ).
The y-component is 220 * sin(180 - θ) = 220 * sin(θ) since sin(180 - θ) = sin(θ), as the sine function is symmetric about 180 degrees.
3) Displacement 3: 280 m at east of north
Similarly, we need to determine the angle between the displacement and the positive x-axis. Since it is east of north, the angle would be 90 degrees plus the angle between the displacement and the negative x-axis.
Let's say the angle between the displacement and the negative x-axis is β. Therefore, the angle between the displacement and the positive x-axis would be 90 + β.
Now we can find the x and y components using trigonometry.
The x-component is 280 * cos(90 + β) = -280 * sin(β) since cos(90 + β) = sin(β) and the sine function is symmetric about 90 degrees.
The y-component is 280 * sin(90 + β) = 280 * cos(β) since sin(90 + β) = cos(β).
Now, let's add up the x and y components to find the overall x and y components:
Overall x-component = -190 m + (-220 * cos(θ)) + (-280 * sin(β))
Overall y-component = 0 m + (220 * sin(θ)) + (280 * cos(β))
Since the spelunker ends up back where she started, the overall x and y components must both be zero:
-190 m + (-220 * cos(θ)) + (-280 * sin(β)) = 0
(220 * sin(θ)) + (280 * cos(β)) = 0
Now we have two equations with two unknowns (θ and β). To solve them, we need additional information or measurements. Without that, we cannot determine the exact values of θ and β, and thus we cannot find the unmeasured displacement and its direction.