While exploring a cave, a spelunker starts at the entrance and moves the following distances. She goes 75.0 m north, 250 m east, 116 m at an angle 30.0° north of east, and 200 m south. Find the resultant displacement from the cave entrance.
X = hor. = 250 + 116cos30 = 350.46m.
Y = ver. = 116sin30 + 75 - 200 = -67m.
tanA = Y / X = -67 / 350.46 = -0.19118,
A = -10.82 deg.,CW = -10.82 + 360 = 349.2 deg.,CCW.
R = X / cosA = = 350.46 / cos349.2 = 357m. @ 349.2 deg.
To find the resultant displacement from the cave entrance, we need to calculate the sum of all the individual displacements.
Given information:
Distance north (d₁) = 75.0 m
Distance east (d₂) = 250 m
Distance at an angle (d₃) = 116 m
Angle (θ) = 30.0° north of east
Distance south (d₄) = 200 m
To solve the problem, we will break down the distances into their horizontal and vertical components.
Horizontal components:
d₂ = Distance east
Horizontal component of d₃ = d₃ * cos(θ)
d₄ = 0 (since it is in the opposite direction)
Vertical components:
d₁ = Distance north
Vertical component of d₃ = d₃ * sin(θ)
d₄ = -Distance south (negative because it is in the opposite direction)
Now, let's calculate the horizontal and vertical components:
Horizontal components:
d₂ = 250 m (Distance east)
Horizontal component of d₃ = 116 m * cos(30.0°) = 100.4619 m
d₄ = 0 m (since it is in the opposite direction)
Vertical components:
d₁ = 75.0 m (Distance north)
Vertical component of d₃ = 116 m * sin(30.0°) = 58 m
d₄ = -200 m (negative because it is in the opposite direction)
Now, let's calculate the sum of the horizontal and vertical components:
Sum of horizontal components = d₂ + (Horizontal component of d₃) + d₄ = 250 m + 100.4619 m + 0 m = 350.4619 m (east)
Sum of vertical components = d₁ + (Vertical component of d₃) + d₄ = 75.0 m + 58 m - 200 m = -67 m (south)
Now, we can use these components to calculate the magnitude and direction of the resultant displacement.
To find the magnitude (R) of the resultant displacement:
R = √((Sum of horizontal components)² + (Sum of vertical components)²)
R = √((350.4619 m)² + (-67 m)²) = √(122853.14 m² + 4489 m²) ≈ 350.8 m
To find the direction (θ) of the resultant displacement:
θ = tan⁻¹((Sum of vertical components) / (Sum of horizontal components))
θ = tan⁻¹((-67 m) / (350.4619 m)) ≈ -11.3°
So, the resultant displacement from the cave entrance is approximately 350.8 m at an angle of -11.3° (south of east).