Calculate Relative Velocity of a Moving Airplane
1) What is the speed of the plane with respect to the ground?
120 m/s due east
2) What is the heading of the plane with respect to the ground? (Let 0° represent due north, 90° represents due east)
43 m/s at an angle of 30° west of due north
3) How far east will the plane travel in 1 hour?
163.7 m/s multiplied by one hour
Answer:
The problem deals with calculating the relative velocity of the plane with respect to the ground using the given velocity of the plane and wind. The speed and heading of the plane relative to the ground can be determined using trigonometric functions and the eastward distance covered by the plane in one hour can be calculated.
This problem involves the principle of relative velocity in two dimensions. Relative velocity is the vector difference of the velocities of two objects. It is observed when one object is in motion relative to the other. First, we need to find the components of the velocity of the wind with respect to the ground. The eastward (x) component of the wind is -43 m/s * cos(30°) and the northward (y) component is -43 m/s * sin(30°).
Next, let's add these components to the velocity of the plane relative to the air, which is flying due east, 120 m/s, to get the plane's velocity relative to the ground. The x component is 120 m/s - 43 m/s * cos(30°) and the y component is 0 m/s - 43 m/s * sin(30°). From these, we can compute the speed of the plane with respect to the ground using the Pythagorean theorem and the heading by calculating the arctangent of the y-component over the x-component.
The distance the plane will travel east in 1 hour can be obtained by multiplying the eastward speed of the plane by 1 hour, resulting in 163.7 m/s multiplied by one hour, giving a distance of 589,320 meters.