As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 57.0° north of west with a speed of 40.0 km/h. (Let i hat represent east and j hat represent north.)

(a) What is the unit-vector expression for the velocity of the hurricane?


It maintains this velocity for 3.50 h, at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.4 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?
km/h i hat + km/h j hat

(c) What is the unit-vector expression for the displacement of the hurricane during the first 3.50 h?
km i hat + km j hat

(d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h?
km i hat + km j hat

(e) How far from Grand Bahama is the eye 5.00 h after it passes over the island?
km

(a) The unit-vector expression for the velocity of the hurricane is given by:

40.0 km/h (cos(57.0°) i hat + sin(57.0°) j hat)

(b) The new velocity of the hurricane is given by:

25.4 km/h (0 i hat + 1 j hat)

(c) The displacement of the hurricane during the first 3.50 h is given by:

3.50 h * 40.0 km/h (cos(57.0°) i hat + sin(57.0°) j hat)

(d) The displacement of the hurricane during the latter 1.50 h is given by:

1.50 h * 25.4 km/h (0 i hat + 1 j hat)

(e) To find the distance from Grand Bahama after 5.00 h, we need to add the displacements from parts (c) and (d) and calculate the magnitude:

|3.50 h * 40.0 km/h (cos(57.0°) i hat + sin(57.0°) j hat) + 1.50 h * 25.4 km/h (0 i hat + 1 j hat)|

(a) To find the unit-vector expression for the velocity of the hurricane, we need to determine the total velocity vector for the given direction and speed.

The initial velocity of the hurricane is moving in a direction 57.0° north of west with a speed of 40.0 km/h.

We can break down this velocity into its north and west components using trigonometry:

The west component = velocity * cos(angle)
The north component = velocity * sin(angle)

Using the given values:
West component = 40.0 km/h * cos(57.0°)
North component = 40.0 km/h * sin(57.0°)

Now, we can express the velocity vector using the i and j unit vectors:

Velocity of the hurricane = west component * i hat + north component * j hat

Substituting the values we found:
Velocity of the hurricane = (40.0 km/h * cos(57.0°)) * i hat + (40.0 km/h * sin(57.0°)) * j hat

(b) After 3.50 hours, the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.4 km/h. This gives us the new velocity.

The new velocity of the hurricane is moving due north with a speed of 25.4 km/h.

To express this as a unit-vector expression, we can write it as:
New velocity of the hurricane = 0 km/h * i hat + 25.4 km/h * j hat

(c) To find the displacement of the hurricane during the first 3.50 hours, we need to multiply the initial velocity by the time.

Displacement = velocity * time

Given that the initial velocity is (40.0 km/h * cos(57.0°)) * i hat + (40.0 km/h * sin(57.0°)) * j hat and the time is 3.50 hours, we multiply both components of the velocity vector by 3.50:

Displacement = (40.0 km/h * cos(57.0°) * 3.50 hours) * i hat + (40.0 km/h * sin(57.0°) * 3.50 hours) * j hat

(d) To find the displacement of the hurricane during the latter 1.50 hours, we use the new velocity.

The displacement is equal to the new velocity multiplied by the time:

Displacement = (0 km/h * 1.50 hours) * i hat + (25.4 km/h * 1.50 hours) * j hat

(e) To find how far from Grand Bahama the eye of the hurricane is 5.00 hours after it passes over the island, we need to calculate the total displacement.

The total displacement is the sum of the displacements during the first 3.50 hours and the latter 1.50 hours.

Total displacement = Displacement during the first 3.50 hours + Displacement during the latter 1.50 hours

Calculate the individual displacements and add them up to find the total displacement:

Total displacement = [ (40.0 km/h * cos(57.0°) * 3.50 hours) + (0 km/h * 1.50 hours) ] * i hat + [ (40.0 km/h * sin(57.0°) * 3.50 hours) + (25.4 km/h * 1.50 hours) ] * j hat

Finally, to find the distance from Grand Bahama, you can use the formula for magnitude of a vector:

Distance = sqrt((total displacement x)^2 + (total displacement y)^2)

(a) To find the unit-vector expression for the velocity of the hurricane, we need to break down its velocity into its east-west (i hat) and north-south (j hat) components.

Given that the hurricane is moving in a direction 57.0° north of west, the east-west component can be found using trigonometry. The sine of the angle can be used to find the north-south component, and the cosine of the angle can be used to find the east-west component.

East-west component = 40.0 km/h * cos(57.0°)
North-south component = 40.0 km/h * sin(57.0°)

The unit-vector expression for velocity can be found by dividing the components by the magnitude of the velocity:

Magnitude = sqrt((east-west component)^2 + (north-south component)^2)

Unit-vector expression = (east-west component / magnitude) i hat + (north-south component / magnitude) j hat

(b) To find the unit-vector expression for the new velocity of the hurricane, we can use the same process as in part (a) with the given values of speed and direction.

The direction is now due north, which means the angle is 90.0°.

East-west component = 25.4 km/h * cos(90.0°)
North-south component = 25.4 km/h * sin(90.0°)

Unit-vector expression = (east-west component / magnitude) i hat + (north-south component / magnitude) j hat

(c) To find the unit-vector expression for the displacement of the hurricane during the first 3.50 h, we need to multiply the initial velocity by the time.

Displacement = initial velocity * time

Initial velocity can be found using the unit-vector expression from part (a).

(d) To find the unit-vector expression for the displacement of the hurricane during the latter 1.50 h, we use the same process as in part (c) but with the new velocity from part (b).

(e) To find how far from Grand Bahama the eye of the hurricane is 5.00 h after it passes over the island, we need to add the displacements from parts (c) and (d) together and calculate the magnitude of the displacement.