An airplane has to fly eastward to a destination 856 km away. If wind is blowing at 18.0 m/s northward and the air speed of the plane is 161 m/s. In what direction should the plane head to reach its destination.
So I know the direction is south east but can someone give me the formula on how to solve the degrees?
To determine the direction in degrees in which the plane should head, we can use trigonometry. We can use the concept of vector addition to find the resultant velocity of the plane, which is the combination of the wind velocity and the air velocity.
Let's break down the given information:
- Wind velocity: 18.0 m/s northward
- Air speed of the plane: 161 m/s (assumed to be relative to the ground)
- Destination distance: 856 km (we will later convert it to meters)
Now, we need to find the resultant velocity vector by adding the vectors of the wind velocity and the air velocity.
1. Convert the destination distance from kilometers to meters:
- 856 km * 1000 m/km = 856,000 m
2. Draw a diagram to visualize the problem:
- Draw a vector representing the air velocity of the plane (161 m/s) from the origin (point A).
- Draw another vector representing the wind velocity (18.0 m/s) from the origin (point A) in the north direction.
3. Use the Pythagorean theorem to find the magnitude of the resultant velocity vector:
- Let's call the resultant velocity vector R.
- R² = (air velocity)² + (wind velocity)²
- R² = (161 m/s)² + (18.0 m/s)²
- R = sqrt[(161 m/s)² + (18.0 m/s)²]
4. Use trigonometry to find the angle of the resultant velocity vector:
- tan(angle) = opposite/adjacent
- y-component of resultant velocity vector (opposite) = wind velocity (18.0 m/s)
- x-component of resultant velocity vector (adjacent) = air velocity of the plane (161 m/s)
- angle = arctan[(wind velocity)/ (air velocity)]
5. Substitute the values and solve for the angle:
- angle = arctan[(18.0 m/s) / (161 m/s)]
- angle = arctan(0.1118)
- angle ≈ 6.4 degrees
Therefore, to reach its destination, the plane should head in the direction approximately 6.4 degrees south of east.