An airplane has an air speed of 112m/s [N 20 degrees W] and has a ground speed of 106 m/s [N 25 degrees W]. Find the velocity of the wind.
To find the velocity of the wind, we need to determine the difference between the airplane's airspeed and ground speed. The wind velocity affects the airplane's ground speed, causing it to deviate from its intended path.
Let's break down the given information:
Airspeed of the airplane = 112 m/s [N 20 degrees W]
Ground speed of the airplane = 106 m/s [N 25 degrees W]
To solve this problem, we'll use vector addition and trigonometry.
Step 1: Resolve the airspeed and ground speed into their north and west components.
For the airspeed:
- The north component = airspeed * sin(20 degrees)
- The west component = airspeed * cos(20 degrees)
For the ground speed:
- The north component = ground speed * sin(25 degrees)
- The west component = ground speed * cos(25 degrees)
Step 2: Calculate the difference between the north and west components of the airspeed and ground speed:
North component difference = north component of ground speed - north component of airspeed
West component difference = west component of ground speed - west component of airspeed
Step 3: Use the Pythagorean theorem to find the magnitude (velocity) of the wind:
Velocity of the wind = √(North component difference^2 + West component difference^2)
Let's plug in the given values and solve the problem.
For the airspeed:
North component = 112 m/s * sin(20 degrees)
= 112 m/s * 0.342 (-0.342 due to "N 20 degrees W" direction)
≈ - 38.46 m/s (south)
West component = 112 m/s * cos(20 degrees)
= 112 m/s * 0.940 (-0.940 due to "N 20 degrees W" direction)
≈ -105.28 m/s (east)
For the ground speed:
North component = 106 m/s * sin(25 degrees)
= 106 m/s * 0.423 (-0.423 due to "N 25 degrees W" direction)
≈ -44.8 m/s (south)
West component = 106 m/s * cos(25 degrees)
= 106 m/s * 0.906 (-0.906 due to "N 25 degrees W" direction)
≈ -96.04 m/s (east)
North component difference = -44.8 m/s - (-38.46 m/s)
= -6.34 m/s (difference in south direction)
West component difference = -96.04 m/s - (-105.28 m/s)
= -9.24 m/s (difference in east direction)
Velocity of the wind = √((-6.34 m/s)^2 + (-9.24 m/s)^2)
≈ √(40.0756 + 85.4736)
≈ √125.5492
≈ 11.20 m/s
Therefore, the velocity of the wind is approximately 11.20 m/s [11.20 m/s south, 11.20 m/s east].