A pilot wishes to fly to a point 450 km due south in 3 hours. A windis blowing from the west at 50 km/hr. By means of a vector diagram, compute the proper heading and speed that the pilot must choose to achieve this objective.
The vector sum of the velocity with respect to air (speed and heading) and the wind velocity vector must be 150 km/s in a south direction.
Write that as a vector equation and solve for the unknown components.
To compute the proper heading and speed for the pilot to achieve the objective, we need to take into account the wind's influence on the plane's movement. We can use vector addition to solve this problem.
1. Start by drawing a coordinate system with x-axis pointing east, y-axis pointing north, and the origin representing the starting point of the plane.
2. Draw a vector representing the desired displacement of the plane. In this case, it will be a vector going 450 km due south. Label it vector A.
3. Next, draw a vector representing the wind's velocity. In this case, it will be a vector going 50 km/hr from west to east (opposite the x-axis). Label it vector B.
4. Now, subtract vector B from vector A to find the resultant vector, which represents the actual velocity that the plane needs to have in order to counteract the wind's effect.
- To do this, we can break down the vectors into their x and y components:
- Vector A has an x-component of 0 km/hr (since it's going purely south) and a y-component of -450 km/hr.
- Vector B has an x-component of 50 km/hr (since it's going from west to east) and a y-component of 0 km/hr.
- Subtracting the corresponding components, we get the resultant vector:
- The x-component of the resultant vector is 0 km/hr - 50 km/hr = -50 km/hr.
- The y-component of the resultant vector is -450 km/hr - 0 km/hr = -450 km/hr.
- Therefore, the resultant vector has an x-component of -50 km/hr and a y-component of -450 km/hr.
5. The resultant vector represents the plane's actual velocity. To find the proper heading and speed, we can calculate the magnitude (speed) and direction of the resultant vector.
- The magnitude of the resultant vector is given by the Pythagorean theorem:
- Magnitude = sqrt((-50 km/hr)^2 + (-450 km/hr)^2) = sqrt(2500 + 202500) = sqrt(205000) ≈ 452.64 km/hr.
- The direction (heading) of the plane can be calculated using trigonometry:
- Direction = arctan((-450 km/hr)/(-50 km/hr)) = arctan(9) ≈ 82.87 degrees.
- Note: Depending on the coordinate system used, the direction may need to be adjusted (e.g., adding or subtracting 180 degrees) to give the proper heading in relation to the compass directions.
6. Therefore, the pilot must choose a heading of approximately 82.87 degrees (adjusted as necessary) and a speed of approximately 452.64 km/hr to achieve the objective of flying 450 km due south in 3 hours, considering the wind's velocity.