Canadian geese migrate essentially along a north-south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one such bird is flying at 100 km/h relative to the air, but there is a 38.0km/h wind blowing from west to east.
a) At what angle relative to the north-south direction should this bird head so that it will be traveling directly southward relative to the ground?
bHow long will it take the bird to cover a ground distance of 600km from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth's magnetic field to fix the north-south direction.)
a) Well, if the bird wants to travel directly southward relative to the ground, it needs to compensate for the wind blowing from west to east. So, it needs to adjust its heading in order for the total velocity to point directly south. We can use trigonometry to find the angle.
Let's call the angle the bird needs to head as θ. The southward velocity relative to the ground would be the bird's airspeed (100 km/h) times the cosine of θ, since cosine gives us the component of velocity in the southward direction.
The eastward velocity due to the wind is 38.0 km/h, and that would be the bird's airspeed times the sine of θ, since sine gives us the component of velocity in the eastward direction.
For the total velocity to be southward, the eastward component due to the wind must cancel out the westward component due to the bird's heading. So, we have:
100 km/h * cos(θ) = 38.0 km/h
Solving for θ:
cos(θ) = 38.0 km/h / 100 km/h
θ = acos(38.0 km/h / 100 km/h)
θ ≈ acos(0.38)
θ ≈ 67.2 degrees
Therefore, the bird should head at an angle of approximately 67.2 degrees relative to the north-south direction to travel directly southward relative to the ground.
b) To find how long it will take the bird to cover a ground distance of 600 km from north to south, we need to divide the ground distance by the southward velocity relative to the ground. The southward velocity relative to the ground is the airspeed (100 km/h) times the cosine of the angle (67.2 degrees) we found in part (a).
Southward velocity relative to the ground = 100 km/h * cos(67.2 degrees)
Time = Distance / Southward velocity relative to the ground
Plugging in the values:
Time = 600 km / (100 km/h * cos(67.2 degrees))
Calculating it gives us:
Time ≈ 15.7 hours
So, it will take the bird approximately 15.7 hours to cover a ground distance of 600 km from north to south. Keep in mind that this calculation assumes the bird maintains a constant speed and direction, without taking into account any potential rest or stops along the way.
a) To determine the angle at which the bird should head, we need to consider the effect of the wind on its flight. Let's denote the angle the bird should head relative to the north-south direction as θ.
Since the wind is blowing from west to east, it will create a horizontal force that opposes the bird's motion and attempts to push it eastward. This means that the bird needs to counteract this force by pointing slightly westward.
To find the angle θ, we can use trigonometry. We know that the bird's ground speed is 100 km/h relative to the air, and the wind speed is 38.0 km/h from west to east. The resulting ground speed will be the vector sum of the bird's airspeed and the wind speed.
Using the concept of vector addition, we can determine that the ground speed is given by:
Ground speed = √(airspeed^2 + wind speed^2)
Plugging in the values, we get:
Ground speed = √(100^2 + 38.0^2) km/h
Ground speed = √(10000 + 1444) km/h
Ground speed = √(11444) km/h
Ground speed ≈ 107.0 km/h
To find the angle θ, we can use the tangent function:
tan(θ) = (wind speed)/(airspeed)
tan(θ) = 38.0/100
θ = tan^(-1)(38.0/100)
θ ≈ 21.8 degrees
Therefore, the bird should head at an angle of approximately 21.8 degrees westward relative to the north-south direction.
b) To determine the time it will take for the bird to cover a ground distance of 600 km from north to south, we need to calculate its effective ground speed.
Since the bird is flying southward relative to the ground, its effective ground speed will be the same as its ground speed.
Using the ground speed calculated earlier, we can determine the time as follows:
Time = Distance / Ground speed
Time = 600 km / 107.0 km/h
Time ≈ 5.61 hours
Therefore, it will take the bird approximately 5.61 hours to cover a ground distance of 600 km from north to south.
a) To determine the angle the bird should head relative to the north-south direction, we can use vector addition. The bird's velocity relative to the ground can be represented as the sum of its velocity relative to the air and the velocity of the wind.
Let's assume the bird's southward velocity relative to the ground is v(g), the bird's velocity relative to the air is v(bird), and the velocity of the wind is v(wind). We can use the Pythagorean theorem to find v(g) as follows:
v(g)^2 = v(bird)^2 + v(wind)^2
Substituting the given values:
v(g) = sqrt((100 km/h)^2 + (38.0 km/h)^2)
Now, since the bird wants to travel directly southward relative to the ground, the angle (θ) it should head relative to the north-south direction can be found using trigonometry:
tan(θ) = v(wind) / v(bird)
Substituting the given values:
θ = arctan(38.0 km/h / 100 km/h)
Using a calculator, we find:
θ ≈ 20.9 degrees
Therefore, the bird should head approximately 20.9 degrees eastward from the north-south direction to travel directly southward relative to the ground.
b) To determine how long it will take the bird to cover a ground distance of 600 km from north to south, we need to calculate the time it takes for the bird to cover that distance based on its ground speed.
The bird's ground speed can be calculated by subtracting the wind's velocity (v(wind)) from the bird's airspeed (v(bird)), as follows:
Ground speed = v(bird) - v(wind)
Substituting the given values:
Ground speed = 100 km/h - 38.0 km/h
Ground speed = 62.0 km/h
Now, we can calculate the time it takes for the bird to cover a ground distance of 600 km using the formula:
Time = Distance / Speed
Substituting the given values:
Time = 600 km / 62.0 km/h
Using a calculator, we find:
Time ≈ 9.68 hours
Therefore, it will take the bird approximately 9.68 hours to cover a ground distance of 600 km from north to south.
a. Tan A = Y/X = -100/38 = -2.63158
A = -69.2o = 69.2o S of E=20.8o E of
S.
The bird must head 20.8o W of S.
b. d = V*t = 600 km
100t = 600
t = 6 h.