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?
a. Vr = -100i + 70km[330o] =
-100i + 60.6-35i = 60.6 - 135i = 148km[65.8o] S. of E = 24.2o
E. of S. = Resultant velocity of the goose.
Do you mean 20o W. of S.? If so,
Direction = 24.2 + 20 = 44.2o W. of S.
b. d = V*t = 1000, t = 1000/V = 1000/148 = 6.76 h.
To answer these questions, we need to break down the bird's velocity into components and calculate the resulting velocity.
Step 1: Decompose the velocity of the bird:
The bird's velocity consists of two components: (1) its velocity relative to the ground and (2) the velocity of the wind.
Given:
- Bird's velocity relative to the air = 100 km/h
- Wind's velocity = 70 km/h at 30 degrees North of West
Now, we can find the components of the bird's velocity:
Bird's velocity component in the North direction (Vbn):
Vbn = Bird's velocity relative to the air * Sin(angle between bird's velocity and North direction)
Vbn = 100 km/h * Sin(20 degrees) [since bird is heading 20 degrees west of north]
Vbn ≈ 34.1 km/h (rounded to one decimal place)
Bird's velocity component in the West direction (Vbw):
Vbw = Bird's velocity relative to the air * Cos(angle between bird's velocity and North direction)
Vbw = 100 km/h * Cos(20 degrees)
Vbw ≈ 93.5 km/h (rounded to one decimal place)
Step 2: Calculate the resultant velocity:
To find the direction the bird should head, we need to consider both the bird's velocity relative to the ground and the wind's velocity. The resultant velocity will be the vector sum of these two velocities.
Resultant velocity component in the North direction (Vgn):
Vgn = Vbn (bird's velocity component in the North direction) + Vwn (wind's velocity component in the North direction)
Vgn = 34.1 km/h + 70 km/h * Sin(30 degrees) [since wind is blowing from 30 degrees North of West]
Vgn ≈ 83.5 km/h (rounded to one decimal place)
Resultant velocity component in the West direction (Vgw):
Vgw = Vbw (bird's velocity component in the West direction) + Vww (wind's velocity component in the West direction)
Vgw = 93.5 km/h + 70 km/h * Cos(30 degrees)
Vgw ≈ 119.5 km/h (rounded to one decimal place)
Therefore, the bird should head in a direction where its velocity component in the North direction is 83.5 km/h, and the velocity component in the West direction is 119.5 km/h.
(a) The bird should head in a direction that is arctan(Vgn/Vgw) relative to the North direction.
∴ Direction = arctan(83.5 km/h / 119.5 km/h)
Direction ≈ 36.6 degrees north of west (rounded to one decimal place)
(b) To calculate the time it will take for the bird to cover a ground distance of 1000 kilometers, we will use the resultant velocity:
Total velocity = sqrt(resultant velocity component in North direction)^2 + resultant velocity component in West direction)^2
Total velocity = sqrt((83.5 km/h)^2 + (119.5 km/h)^2)
Total velocity ≈ 146.3 km/h (rounded to one decimal place)
Time = Distance / Speed
Time = 1000 km / 146.3 km/h
Time ≈ 6.8 hours (rounded to one decimal place)
Therefore, the bird will take approximately 6.8 hours to cover a ground distance of 1000 kilometers.