Question

An airplane pilot checks her instrument and finds that the speed of the plane relative to the air is 325km/h. the instrument also show that the plane is pointed in a direction [s30w]. a radio report indicate that the wind velocity is 80 km/h [w]. what is the velocity of the plane relative to the groundas it is recorded by an air trffic controlller in a nearby airport?

Answer 1

To find the plane's groundspeed, we decompose the plane's velocity and wind velocity into components, sum those components, and then calculate magnitude and direction of the resultant vector.

Step-by-step explanation:

To determine the velocity of the plane relative to the ground as recorded by an air traffic controller at a nearby airport, we need to perform a vector addition of the plane's velocity relative to the air and the wind velocity. The plane's speed relative to the air is given as 325 km/h in a direction of [S30W]. The wind velocity is given as 80 km/h [W].

We can break these velocities into components using trigonometric functions. The [S30W] direction can be decomposed into a southward and westward component, and since it forms a 30-degree angle to the south of west, the components will be:

• Southward: 325 km/h * sin(30°)
• Westward: 325 km/h * cos(30°)

The wind velocity has only a westward component, which is -80 km/h; the negative sign indicates that it opposes the eastward direction.

Now we can sum the components to find the resultant velocity relative to the ground:

• RESULTANT_South: 325 km/h * sin(30°)
• RESULTANT_West: 325 km/h * cos(30°) - 80 km/h

Finally, we can calculate the magnitude and direction of the resultant velocity vector using the Pythagorean theorem and trigonometry to find the groundspeed and direction.