An airplane is flying North with a velocity of 22 m/s. A strong wind is blowing East at 50 m/s. What is the airplane's resultant velocity (magnitude and direction)?
Add the vectors. The air speed and wind speed are perpendicular. The resultant speed is the hypotenuse of the vector right triangle. Draw the triangle and you will see that the hypotenuse is in the direction of the resultant velocity.
Are you sure about the 22 m/s airplane speed? Few planes fly that slow. It's only about 50 mph.
I am willing to bet the airplane speed is supposed to be 220 m/s
To find the airplane's resultant velocity, we need to consider the vector sum of the airplane's velocity and the wind's velocity.
Step 1: Represent the airplane's velocity as a vector. Since the airplane is flying North, its velocity can be represented as a vector pointing directly upwards. Let's call this vector V_airplane.
Step 2: Represent the wind's velocity as a vector. Since the wind is blowing East, its velocity can be represented as a vector pointing to the right. Let's call this vector V_wind.
Step 3: Add the two vectors together to find the resultant velocity. To do this, we can use vector addition. Add the magnitudes of the vectors and take into account their directions.
The magnitude of the airplane's velocity is given as 22 m/s, and the magnitude of the wind's velocity is given as 50 m/s. The directions are at right angles to each other, so we can use the Pythagorean theorem to find the magnitude of the resultant velocity.
Magnitude of the resultant velocity (V_resultant) = sqrt((magnitude of V_airplane)^2 + (magnitude of V_wind)^2)
Plugging in the values, we get:
V_resultant = sqrt((22 m/s)^2 + (50 m/s)^2)
V_resultant = sqrt(484 m^2/s^2 + 2500 m^2/s^2)
V_resultant = sqrt(2984 m^2/s^2)
V_resultant ≈ 54.65 m/s
So, the magnitude of the airplane's resultant velocity is approximately 54.65 m/s.
To find the direction, we can calculate the angle between the resultant vector and the reference direction (North). We can use the inverse tangent function.
Direction of the resultant velocity = atan((magnitude of V_wind) / (magnitude of V_airplane))
Plugging in the values, we get:
Direction of the resultant velocity = atan(50 m/s / 22 m/s)
Direction of the resultant velocity ≈ 1.143 radians
Converting to degrees, we get:
Direction of the resultant velocity ≈ 1.143 radians * (180 degrees / pi radians)
Direction of the resultant velocity ≈ 65.54 degrees
Therefore, the airplane's resultant velocity is approximately 54.65 m/s in a direction of 65.54 degrees North of East.