The eye of a hurricane passes over Grand Bahama Island. It is moving in a direction θ = 50° north of west with speed v1 = 34.5 km/h. Exactly three hours later, the course of the hurricane shifts due north, and its speed slows to v2 = 16.1 km/h, as shown below. How far from Grand Bahama is the hurricane 3.5 h after it passes over the island?
To solve this problem, we need to break it down into steps and use vector addition.
Step 1: Determine the displacement of the hurricane during the first 3 hours.
Since the hurricane is moving at a speed of v1 = 34.5 km/h for 3 hours and in a direction θ = 50° north of west, we can find its displacement in that time using the formula:
Displacement = Speed × Time
Displacement = v1 × 3
Displacement = 34.5 km/h × 3 h
Displacement = 103.5 km
Step 2: Determine the displacement of the hurricane during the additional 0.5 hours.
Since the course of the hurricane shifts due north and its speed slows to v2 = 16.1 km/h, we can find the additional displacement using the same formula as in step 1:
Displacement = Speed × Time
Displacement = v2 × 0.5
Displacement = 16.1 km/h × 0.5 h
Displacement = 8.05 km
Step 3: Determine the magnitude and direction of the total displacement of the hurricane.
To find the magnitude of the total displacement, we need to add the displacements found in steps 1 and 2. To find the direction, we can use trigonometry.
Magnitude of total displacement = √(Displacement1^2 + Displacement2^2)
Magnitude of total displacement = √(103.5 km^2 + 8.05 km^2)
Magnitude of total displacement = √(10702.45 km^2)
Magnitude of total displacement ≈ 103.45 km
Direction of total displacement = arctan(Displacement2 / Displacement1)
Direction of total displacement = arctan(8.05 km / 103.5 km)
Direction of total displacement ≈ 4.45°
Step 4: Determine the final distance from Grand Bahama Island after 3.5 hours.
To find the final distance, we can use the magnitude of the total displacement and the Pythagorean theorem.
Final distance = √((Magnitude of total displacement^2) + (Distance traveled north^2))
Final distance = √((103.45 km^2) + (3.5 h * v2)^2)
Final distance = √((10702.45 km^2) + (3.5 h * 16.1 km/h)^2)
Final distance ≈ √(10702.45 km^2 + (3.5 h * 16.1 km/h)^2)
Final distance ≈ √(10702.45 km^2 + (3.5 h * 16.1 km/h)^2)
Final distance ≈ √(10702.45 km^2 + (3.5 * 16.1)^2 km^2)
Final distance ≈ √(10702.45 + 899.8)
Final distance ≈ √(11502.25)
Final distance ≈ 107.24 km
Therefore, the hurricane is approximately 107.24 km away from Grand Bahama Island 3.5 hours after it passes over the island.