An airplane heads due south with an air speed of 480km/h. Measurements made from the ground indicate that the plane's ground speed is 528km/h at 15 degrees east of south. Calculate the wind speed.
make a sketch with O the origin
OP = 480 and south along the y-axis.
draw OR as the resultant of length 528 and angle POR = 15°
Complete the parallogram with OQ =PR and QR = OP
by cosine law:
PR^2 = 480^2 + 528^2 + 2(480)(528)cos15°
PR = 139.9125
So the wind speed is 139.9 km/h
so by using that formula for cosine i somehow still can't get the right answer :(
To find the wind speed, we need to calculate the difference between the airplane's airspeed and ground speed.
Step 1: Draw a diagram to visualize the problem.
Let's draw a vector diagram to represent the airplane's velocity relative to the ground and the wind velocity.
The airplane's airspeed is due south (downward direction), and the ground speed is at 15 degrees east of south.
Label the airspeed vector as "A" and the ground speed vector as "G".
A
/
/
/
───┼─── G
\
\
\
\
Step 2: Find the component of the ground speed vector along the south direction.
Given that the magnitude of the ground speed is 528 km/h, we can find the component of the ground speed vector along the south direction by multiplying the ground speed by the sine of the angle (15 degrees).
Component in the south direction = Ground speed * sin(angle)
Component in the south direction = 528 km/h * sin(15°)
Step 3: Calculate the component of the wind speed vector along the south direction.
Since the airspeed is due south, the wind speed vector is perpendicular to the airspeed vector.
The component of the wind speed vector along the south direction is equal to the component of the ground speed vector along the south direction.
Component of the wind speed vector along the south direction = Component in the south direction = 528 km/h * sin(15°)
Step 4: Find the magnitude of the wind speed.
The wind speed vector can be found by taking the difference between the airspeed vector and the ground speed vector.
Magnitude of the wind speed = Magnitude of the airspeed - Component of the wind speed vector along the south direction
Magnitude of the wind speed = 480 km/h - Component of the wind speed vector along the south direction
Magnitude of the wind speed = 480 km/h - (528 km/h * sin(15°))
Now, we can calculate the wind speed by substituting the given values into the equation:
Magnitude of the wind speed = 480 km/h - (528 km/h * sin(15°))
Calculating this value, we find the magnitude of the wind speed to be approximately 64.5 km/h.
Therefore, the wind speed is 64.5 km/h.
To calculate the wind speed, we need to use vector addition.
Let's break down the airplane's velocity into two components: the actual velocity of the airplane in still air and the velocity of the wind.
The actual velocity of the airplane can be represented as a vector pointing due south with a magnitude of 480 km/h.
The velocity of the wind can be represented as a vector that is added to the airplane's velocity vector to give us the ground speed of 528 km/h at 15 degrees east of south.
To find the wind speed, we can draw a diagram to help visualize the situation:
|
/|\
/ | \
/ | \
/ θ| \
/ | \
/ | \
/______|______\
\
In this diagram, the length of the line represents the magnitude of the vectors, and the angle θ represents the 15 degrees east of south. We can label the length of the airplane's velocity vector as 480 km/h and the length of the wind velocity vector as w km/h (the unknown we are trying to find).
Since the ground speed is the resultant vector of the airplane's velocity and the wind velocity, we can use the law of cosines to find the magnitude of the wind velocity vector:
ground speed² = airplane velocity² + wind velocity² - 2 * airplane velocity * wind velocity * cos(θ)
Substituting the given values, we have:
528 km/h² = 480 km/h² + w km/h² - 2 * 480 km/h * w km/h * cos(15)
Simplifying the equation:
528² = 480² + w² - 960w * cos(15)
Solving for w, the wind speed:
w² - 960w * cos(15) + (480² - 528²) = 0
Now we can solve this quadratic equation to find the wind speed using either factoring, the quadratic formula, or any other suitable method.
Hope this helps!