A jet is flying through a wind that is blowing with a speed of 45 mi/h in the direction N 30° E (see the figure). The jet has a speed of 775 mi/h relative to the air, and the pilot heads the jet in the direction N 45° E.
To determine the resultant velocity of the jet, we need to find the vector sum of its velocity relative to the air and the velocity of the wind. Here's how you can calculate it step by step:
1. Convert the given velocities to vector form:
- The wind velocity is 45 mi/h in the direction N 30° E. We can break it down into its northward (N) and eastward (E) components:
- The northward component = 45 mi/h * sin(30°) = 22.5 mi/h (opposite to southward)
- The eastward component = 45 mi/h * cos(30°) = 38.9 mi/h (eastward)
- The jet's velocity relative to the air is 775 mi/h in the direction N 45° E. Again, we can determine its components as follows:
- The northward component = 775 mi/h * sin(45°) = 547.7 mi/h (opposite to southward)
- The eastward component = 775 mi/h * cos(45°) = 547.7 mi/h (eastward)
2. Add the components together to find the resultant velocity:
- Combine the northward components:
- Jet's northward velocity: -547.7 mi/h + (-22.5 mi/h) = -570.2 mi/h (opposite to southward)
- Combine the eastward components:
- Jet's eastward velocity: 547.7 mi/h + 38.9 mi/h = 586.6 mi/h (eastward)
3. Convert the resultant velocity back to magnitude and direction:
- To find the magnitude (speed) of the resultant velocity, use the Pythagorean theorem:
- Magnitude = sqrt((-570.2 mi/h)^2 + (586.6 mi/h)^2) ≈ 815.6 mi/h
- To find the direction, use the inverse tangent function:
- Direction = atan((-570.2 mi/h) / (586.6 mi/h)) ≈ -45.5° (west of north)
So, the resultant velocity of the jet, considering both its velocity relative to the air and the wind, is approximately 815.6 mi/h in the direction N 45.5° W.