The eye of a hurricane passes over Grand Bahama Island. It is moving in a direction 60.0° north of west with a speed of v1 = 42.0 km/h. Exactly three hours later, the course of the hurricane shifts due north, and its speed slows to v2 = 24.0 km/h, as shown in Figure 3-30. How far from Grand Bahama, in kilometers, is the hurricane 4.50 h after it passes over the island?
To find the distance from Grand Bahama Island to the hurricane 4.50 hours after it passes over the island, we need to break down the movement of the hurricane into its north and west components.
Let's divide the problem into two parts:
Part 1: The first three hours
During the first three hours, the hurricane is moving at a speed of v1 = 42.0 km/h in a direction 60.0° north of west. We need to find the distance it traveled during this time.
To find the westward component of the velocity, we can use the formula: v1west = v1 * cos(60.0°).
v1west = 42.0 km/h * cos(60.0°) = 21.0 km/h
To find the northward component of the velocity, we can use the formula: v1north = v1 * sin(60.0°).
v1north = 42.0 km/h * sin(60.0°) = 36.4 km/h
During the first three hours, the distance traveled westward is given by: distance_west = v1west * time = 21.0 km/h * 3 h = 63.0 km.
The distance traveled northward is given by: distance_north = v1north * time = 36.4 km/h * 3 h = 109.2 km.
Part 2: After the course shift
After the course shift, the hurricane moves north at a speed of v2 = 24.0 km/h. We need to find the additional distance it travels north in the remaining time.
The distance traveled north during the remaining time can be calculated as: distance_north_remaining = v2 * time_remaining.
The remaining time is given by: time_remaining = total_time - time_part_1 = 4.50 h - 3 h = 1.50 h.
Substituting the values, we get: distance_north_remaining = 24.0 km/h * 1.50 h = 36.0 km.
Finally, to find the total distance from Grand Bahama Island to the hurricane after 4.50 hours, we add the distances traveled westward, northward, and the additional distance traveled northward: total_distance = distance_west + distance_north + distance_north_remaining.
Substituting the previously calculated values, we get: total_distance = 63.0 km + 109.2 km + 36.0 km = 208.2 km.
Therefore, the hurricane is approximately 208.2 kilometers away from Grand Bahama Island after 4.50 hours.