An airplane is flying south at 500 km/hr and is being blown by a crosswind of 75 km/hr east. What is the resultant velocity of the airplane?
the angle E of S is arc tan 75/500. determine that angle
sintheta=75/speed
solve for speed
To find the resultant velocity of the airplane, we can use vector addition. The airplane's velocity can be represented by a vector pointing south with a magnitude of 500 km/hr, and the crosswind can be represented by a vector pointing east with a magnitude of 75 km/hr.
To add these vectors, we can use the Pythagorean theorem and trigonometry. Let's consider the horizontal and vertical components of the vectors.
The horizontal component of the airplane's velocity is its speed multiplied by the cosine of the angle between its velocity vector and the horizontal axis. Since the airplane is flying directly south, the cosine of this angle is 0. Therefore, the horizontal component of the airplane's velocity is 0 km/hr.
The horizontal component of the crosswind is its speed multiplied by the cosine of the angle between its velocity vector and the horizontal axis. Since the crosswind is blowing directly east, this angle is 0 degrees and the cosine of 0 degrees is 1. Therefore, the horizontal component of the crosswind is 75 km/hr.
The vertical component of the airplane's velocity is its speed multiplied by the sine of the angle between its velocity vector and the horizontal axis. Since the airplane is flying directly south, the sine of this angle is 1. Therefore, the vertical component of the airplane's velocity is 500 km/hr.
The vertical component of the crosswind is its speed multiplied by the sine of the angle between its velocity vector and the horizontal axis. Since the crosswind is blowing directly east, this angle is 90 degrees and the sine of 90 degrees is 1. Therefore, the vertical component of the crosswind is 0 km/hr.
Now, we can add the horizontal and vertical components of the airplane's velocity and the crosswind's velocity separately to get the resultant velocity:
Horizontal component: 0 km/hr + 75 km/hr = 75 km/hr
Vertical component: 500 km/hr + 0 km/hr = 500 km/hr
Using the Pythagorean theorem, we can find the magnitude of the resultant velocity:
Resultant velocity = sqrt((horizontal component)^2 + (vertical component)^2)
Resultant velocity = sqrt((75 km/hr)^2 + (500 km/hr)^2)
Resultant velocity ≈ 505.04 km/hr
Therefore, the resultant velocity of the airplane is approximately 505.04 km/hr.