Canadian snowbirds usually follow the I-15 at a speed of about 120 kilometers per hour. Canadian
geese, on the other hand, migrate approximately along a north-south direction for well over a thousand
kilometers in some cases, traveling at speeds up to about 100 kilometers per hour. Suppose one such
bird (Candian goose) is flying at 100 kilometers per hour relative to the air, but there is a 70 kilometer
per hour wind blowing from 30 degrees north of west,
(a) at what direction should the bird head so that it will be traveling 20 degrees west of north relative
to the ground?
(b) How long will it take the bird to cover a ground distance of 1000 kilometers?
See previous post: Mon, 2-8-16, 7:33 PM.
Edward travels 150 kilometers due west and then 200 kilometers in a direction 60° north of west. What is his displacement in the westerly direction?
To determine the displacement in the westerly direction, we need to find the x-component of the displacement vector.
First, let's analyze the given distances traveled:
* Edward travels 150 kilometers due west. This means he has traveled 150 kilometers in the negative x-direction.
* Edward then travels 200 kilometers in a direction 60° north of west. To find the x-component of this distance, we'll use trigonometry. The angle between the north direction and the west direction is 90°, so the angle between the north direction and the direction Edward travels is 90° - 60° = 30°. The x-component of this distance can be found using cosine:
x-component = 200 km * cos(30°) ≈ 200 km * 0.866 = 173.2 km
Now, let's calculate the displacement in the westerly direction:
The displacement in the x-direction is the sum of the x-components of the distances traveled:
Displacement in the x-direction = -150 km + 173.2 km = 23.2 km
Therefore, Edward's displacement in the westerly direction is approximately 23.2 kilometers.
To solve this problem, we can break it down into two components: the bird's velocity relative to the wind, and the bird's resulting velocity relative to the ground.
(a) To determine the direction the bird should head, we need to find the resultant velocity vector by subtracting the wind velocity from the bird's air velocity. We can do this by using vector addition. Here's how:
Step 1: Resolve the wind velocity vector into its north and west components:
- The wind speed is given as 70 km/h at an angle of 30 degrees north of west.
- The westward component of the wind velocity is given by 70 km/h * cos(30 degrees).
- The northward component of the wind velocity is given by 70 km/h * sin(30 degrees).
Step 2: Subtract the wind vector components from the bird's air velocity vector:
- The bird's air velocity is given as 100 km/h.
- The bird's northward velocity relative to the ground is the difference between the bird's air velocity and the northward component of the wind velocity.
- The bird's westward velocity relative to the ground is the sum of the bird's air velocity component and the westward component of the wind velocity.
Step 3: Use trigonometry to calculate the resulting velocity's direction relative to the ground:
- The direction angle of the resultant velocity vector relative to the ground is given by:
- tan(angle) = (northward velocity)/(westward velocity)
- angle = arctan((northward velocity)/(westward velocity))
(b) To find the time it will take for the bird to cover a ground distance of 1000 kilometers, we can divide the total distance by the ground velocity obtained from part (a).
Let's calculate:
(a)
Step 1: Resolve the wind velocity vector:
- Westward component = 70 km/h * cos(30 degrees) ≈ 60.62 km/h
- Northward component = 70 km/h * sin(30 degrees) ≈ 35 km/h
Step 2: Find the bird's velocity relative to the ground:
- Northward velocity relative to the ground = 100 km/h - 35 km/h = 65 km/h
- Westward velocity relative to the ground = 60.62 km/h
Step 3: Calculate the direction angle relative to the ground:
- tan(angle) = (65 km/h)/(60.62 km/h)
- angle = arctan(65/60.62) ≈ 46.33 degrees
Therefore, the bird should head at approximately 20 degrees west of north relative to the ground.
(b) To find the time, divide the distance by the ground velocity:
- Time = Distance / Ground velocity
- Time = 1000 km / 65 km/h ≈ 15.4 hours
Therefore, it will take the bird approximately 15.4 hours to cover a ground distance of 1000 kilometers.