As it passes over Grand Bahama Island, the eye of a hurricane is moving in a dir
ID: 2047733 • Letter: A
Question
As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 65.0° north of west with a speed of 40.8 km/h. (Let i hat bold represent east and j hat bold represent north.)(a) What is the unit-vector expression for the velocity of the hurricane?
km/h ihatbold + km/h jhatbold
It maintains this velocity for 3.40 h, at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 22.8 km/h. This new velocity is maintained for 1.50 h.
(b) What is the unit-vector expression for the new velocity of the hurricane?
(c) What is the unit-vector expression for the displacement of the hurricane during the first 3.40 h?
(d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h?
(e) How far from Grand Bahama is the eye 4.90 h after it passes over the island?
km
Explanation / Answer
Start with Grand Bahama at (0,0). 60 degrees north of west is a line (call it line A) 60 degrees up from the X axis pointing to the left or 30 degrees left of the y axis. The hurricane travels for three hours along this line at 41 km/h or a total of 123 km. Now, from that point, due north is a line (line B) parallel to the y axis going up. It travels for 1.5h at 25 km/h or 37.5 km. You now have two lines end-to-end for a total travel time of 4.5 hrs. The total distance from the Grand Bahama is the length of the line you draw from (0,0) to the end of line B. If you draw a line along the y axis from (0,0) stopping at the same height as the end point of line B then a line from there to the end point of line B, you have completed a 30 degree right-triangle. To help figure out the total distance, first draw a line from the end of line A, parallel to the x axis, ending at the y axis. This makes a smaller right triangle. The short leg of the right triangle is 123km X sin(30 degrees) or 61.5km, equal to the short leg of the larger triangle. The long leg of the smaller triangle is 123km X cos(30 degrees) or 106.5km. This result added to line B, 37.5km, equals 144km, the long leg of the larger triangle. By taking the square of 61.5 and 144, adding together and taking the square root to find the hypotenuse, you get 156.6km, the total distance the hurricane has traveled from Grand Bahama Island in 4.5 hours.
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