A bird is flying 30 mph in a direction 30 degrees south of east. Affecting the flight of the bird is a wind blowing from the northeast to the southwest at a speed of 10 mph. What is the resulting speed of the bird?
draw the vector triangle (head to tail).
I have on my sketch 30 at 30deg SofE, and then on its head a vector of 10 pointing SW. The angle (check this) between the two vectors is (30+45) degrees
Law of cosines>
R^2=10^2+30^2 - 2*10*30*cos75
solve for the resultant R
26
To find the resulting speed of the bird, we can use vector addition.
First, we need to convert the given information into vectors.
The bird's velocity can be represented as a vector with magnitude 30 mph and direction 30 degrees south of east. We can break this velocity into its horizontal and vertical components using trigonometry.
The horizontal component is given by:
cos(30 degrees) * 30 mph = (sqrt(3)/2) * 30 mph = 15 * sqrt(3) mph
The vertical component is given by:
sin(30 degrees) * 30 mph = (1/2) * 30 mph = 15 mph
So, the bird's velocity is 15 * sqrt(3) mph to the right (east) and 15 mph downward (south).
Next, we need to represent the wind velocity as a vector. The wind is blowing from the northeast to the southwest, which is 45 degrees south of east. The speed of the wind is 10 mph.
The wind's velocity can be broken into its horizontal and vertical components using trigonometry.
The horizontal component is given by:
cos(45 degrees) * 10 mph = (sqrt(2)/2) * 10 mph = 5 * sqrt(2) mph
The vertical component is given by:
sin(45 degrees) * 10 mph = (sqrt(2)/2) * 10 mph = 5 * sqrt(2) mph
So, the wind's velocity is 5 * sqrt(2) mph to the right (east) and 5 * sqrt(2) mph downward (south).
Finally, we can find the resulting speed of the bird by adding the horizontal and vertical components of the bird's and wind's velocities.
The resulting horizontal component is:
(15 * sqrt(3) mph) + (5 * sqrt(2) mph) = 15 * sqrt(3) + 5 * sqrt(2) mph
The resulting vertical component is:
(15 mph) - (5 * sqrt(2) mph) = 15 mph - 5 * sqrt(2) mph
Using the Pythagorean theorem, we can find the magnitude (resulting speed) of the resulting velocity.
Magnitude of the resulting velocity = sqrt((resulting horizontal component)^2 + (resulting vertical component)^2)
Magnitude of the resulting velocity = sqrt((15 * sqrt(3) + 5 * sqrt(2))^2 + (15 - 5 * sqrt(2))^2)
Simplifying this expression will give the resulting speed of the bird.
To determine the resulting speed of the bird, we need to combine the bird's velocity with the velocity of the wind.
First, we need to break down the bird's velocity into its horizontal and vertical components. We can find the horizontal component by multiplying the bird's speed by the cosine of the angle between its direction and the east direction.
Horizontal Component = Bird's Speed * cosine(Angle)
In this case, the bird's speed is 30 mph, and the angle is 30 degrees south of east. So we have:
Horizontal Component = 30 mph * cosine(30 degrees) = 30 mph * 0.866 = 25.98 mph (rounded to two decimal places)
The vertical component can be found by multiplying the bird's speed by the sine of the angle between its direction and the east direction.
Vertical Component = Bird's Speed * sine(Angle)
Using the same values as before, we get:
Vertical Component = 30 mph * sine(30 degrees) = 30 mph * 0.5 = 15 mph
Now, we can consider the effect of the wind. The wind is blowing from the northeast to the southwest, which is perpendicular to the bird's flight direction. We can think of the wind's velocity as a vector with a horizontal and vertical component.
The horizontal component of the wind is the same as its speed, which is 10 mph. However, since it's blowing from the northeast, its horizontal component will be negative.
Horizontal Component of Wind = -10 mph
The vertical component of the wind is also -10 mph since it's blowing from the northeast to the southwest.
Vertical Component of Wind = -10 mph
To determine the resulting speed of the bird, we need to add the horizontal and vertical components of the bird's velocity to the horizontal and vertical components of the wind's velocity.
Horizontal Resultant Velocity = Horizontal Component of Bird's Velocity + Horizontal Component of Wind
Vertical Resultant Velocity = Vertical Component of Bird's Velocity + Vertical Component of Wind
In this case, the bird's velocity is solely in the horizontal plane, so the vertical component of the bird's velocity and the vertical component of the wind's velocity are both zero.
Horizontal Resultant Velocity = 25.98 mph + (-10 mph) = 15.98 mph (rounded to two decimal places)
Finally, we can find the resulting speed of the bird by using the Pythagorean theorem:
Resultant Speed = square root((Horizontal Resultant Velocity)^2 + (Vertical Resultant Velocity)^2)
In this case, since the vertical component is zero, the resulting speed simplifies to:
Resultant Speed = square root((15.98 mph)^2 + (0 mph)^2) = square root(255.68 mph^2) = 15.99 mph (rounded to two decimal places)
Therefore, the resulting speed of the bird is approximately 15.99 mph.