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?
To solve this problem, we can break it down into two components: the horizontal component and the vertical component. The horizontal component represents the bird's movement parallel to the ground, and the vertical component represents the bird's movement perpendicular to the ground.
(a) To determine the direction the bird should head, we need to find the resultant velocity by combining the bird's velocity relative to the air and the velocity of the wind.
First, let's find the horizontal and vertical velocities of the bird relative to the air:
Horizontal velocity of the bird = Velocity of the bird * cos(20°)
= 100 km/h * cos(20°)
Vertical velocity of the bird = Velocity of the bird * sin(20°)
= 100 km/h * sin(20°)
Now, let's find the horizontal and vertical velocities of the wind:
Horizontal velocity of the wind = Velocity of the wind * cos(30°)
= 70 km/h * cos(30°)
Vertical velocity of the wind = Velocity of the wind * sin(30°)
= 70 km/h * sin(30°)
Next, we can find the resultant horizontal and vertical velocities by adding the respective velocities together:
Resultant horizontal velocity = Horizontal velocity of the bird + Horizontal velocity of the wind
= 100 km/h * cos(20°) + 70 km/h * cos(30°)
Resultant vertical velocity = Vertical velocity of the bird + Vertical velocity of the wind
= 100 km/h * sin(20°) + 70 km/h * sin(30°)
Now, we have the resultant velocity, which can be calculated using the Pythagorean theorem:
Resultant velocity = √((Resultant horizontal velocity)^2 + (Resultant vertical velocity)^2)
Finally, we can find the direction of the bird's movement relative to the ground:
Direction = arctan(Resultant vertical velocity / Resultant horizontal velocity) + 270°
(b) To calculate the time it will take for the bird to cover a ground distance of 1000 kilometers, we can use the formula:
Time = Distance / Ground velocity
The ground velocity is the resultant velocity calculated above.
Note: The calculations involve trigonometric functions and may require a calculator or software for precise values and conversions.