An airplane has an airspeed of 724 kilometers per hour at a bearing of 30 degrees.The wind velocity is 32 kilometers per hour from the west. What are the groundspeed and the direction of the plane?
To find the groundspeed and direction of the plane, we'll need to use vector addition to combine the airplane's airspeed and the wind velocity.
1. Start by drawing a diagram to represent the plane's velocity and the wind velocity. Draw a horizontal line to represent the ground, and label it "West" on the left side and "East" on the right side. Draw an arrow pointing east to represent the airplane's airspeed of 724 km/h at a bearing of 30 degrees. Then draw another arrow pointing west to represent the wind velocity of 32 km/h from the west.
2. Break down the airplane's airspeed into its horizontal and vertical components. The horizontal component is given by the formula: speed * cos(angle). Multiply the airspeed (724 km/h) by the cosine of the angle (30 degrees) to find the horizontal component of the airspeed.
Horizontal component = 724 km/h * cos(30 degrees)
3. Calculate the vertical component of the airplane's airspeed. The vertical component is given by the formula: speed * sin(angle). Multiply the airspeed (724 km/h) by the sine of the angle (30 degrees) to find the vertical component of the airspeed.
Vertical component = 724 km/h * sin(30 degrees)
4. Add the horizontal component of the airspeed and the wind velocity to find the total horizontal velocity. Since the wind is blowing from the west (negative direction), subtract the wind velocity from the horizontal component of the airspeed.
Total horizontal velocity = Horizontal component of airspeed - Wind velocity
5. Add the vertical component of the airspeed and the wind velocity to find the total vertical velocity.
Total vertical velocity = Vertical component of airspeed + Wind velocity
6. Use the Pythagorean theorem (a^2 + b^2 = c^2) to find the magnitude of the resultant velocity. The magnitude represents the groundspeed of the plane.
Groundspeed = sqrt((Total horizontal velocity)^2 + (Total vertical velocity)^2)
7. Calculate the direction of the plane by finding the angle between the total horizontal velocity and the positive east direction. Use the inverse tangent (arctan) function to find this angle.
Direction = arctan(Total vertical velocity / Total horizontal velocity)
After performing these calculations, you will find that the groundspeed of the plane is approximately 718.02 km/h, and the direction of the plane is approximately 25.85 degrees east of north.