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?
I just can't find out what direction it needs to go.
This is a vector addition problem. Break into components.
To find the direction the Canadian goose should head in order to be traveling 20 degrees west of north relative to the ground, we need to consider the vector addition of the bird's velocity relative to the air and the velocity of the wind.
Here's how we can solve it step by step:
Step 1: Represent the velocity of the bird relative to the air as a vector. Let's call it V(bird-air). It has a magnitude of 100 kilometers per hour and a direction of 0 degrees (north).
Step 2: Represent the velocity of the wind as a vector. Let's call it V(wind). It has a magnitude of 70 kilometers per hour and a direction of 30 degrees north of west.
Step 3: Find the resultant velocity vector by adding V(bird-air) and V(wind). To do this, we need to break down both vectors into their horizontal and vertical components.
Step 4: Determine the horizontal and vertical components of V(bird-air) and V(wind). Since V(bird-air) is directed north, its horizontal component is 0 and its vertical component is 100 kilometers per hour. For V(wind), we need to find its horizontal and vertical components using trigonometry. The horizontal component of V(wind) is 70 * cos(30°) and the vertical component is 70 * sin(30°).
Step 5: Add the horizontal and vertical components of V(bird-air) and V(wind) to find the resultant horizontal and vertical components.
Step 6: Determine the magnitude and direction of the resultant velocity vector. The magnitude is found by taking the square root of the sum of the squares of the resultant horizontal and vertical components. The direction is found using the arctangent function (inverse tangent) of the quotient of the resultant horizontal and vertical components.
Step 7: Adjust the direction of the resultant velocity vector by 20 degrees west of north.
Step 8: The final direction the bird should head will be the adjusted direction of the resultant velocity vector.
By following these steps, you should be able to determine the direction the bird should head to be traveling 20 degrees west of north relative to the ground.