Navigating Wind: Solving Plane and Escalator Problems

• iJamJL
In summary: I know I shouldn't be double-posting, but I just wanted to bump it so that people know I edited the post. I saw a mistake, and now it better reflects where I'm stuck at.Hint for elevator problem: set the compasses to a perpendicular line.
iJamJL

Homework Statement

Assume all speeds are constant.

(a) A plane flies at speed vstill = 203 km/h in still air. Now, there is a wind blowing at speed vw = 77.6 km/h, with its direction at 30.4 degrees to the east of north. If the pilot wishes to fly due north with the wind blowing, find:
- θ, the angle between the direction the plane flies and due north.

- vp, the speed at which the plane will fly relative to the ground.
(b) A man walks up a stalled escalator in time tw = 89.3 s. If he stands on the moving escalator, he reaches to top in time tm = 26.4 s. Find twm, the time it would take him to reach the top if he walked up the moving escalator?

Homework Equations

The only one I think that is possible is the Law of Cosines.

The Attempt at a Solution

I have one vector going directly North because that is what we began with. Then there is a Northeast wind blowing, which is 30.4°. I put this vector's tail at the tail of the first vector going North. The resultant vector is then 141.8 if we use the Law of Cosines. As for the angle that the plane must fly, I'm at a loss, and have no idea. You don't necessarily have to tell me the answer, but where do I begin?

For the second question, I just have no idea at all.

Last edited:

I know I shouldn't be double-posting, but I just wanted to bump it so that people know I edited the post. I saw a mistake, and now it better reflects where I'm stuck at.

Hint for elevator problem:

Write expressions for the velocity of man walking up steps to top and also for elevator taking him to top. What is velocity if he walks up moving steps?

iJamJL said:

Homework Statement

Assume all speeds are constant.

(a) A plane flies at speed vstill = 203 km/h in still air. Now, there is a wind blowing at speed vw = 77.6 km/h, with its direction at 30.4 degrees to the east of north. If the pilot wishes to fly due north with the wind blowing, find:
- θ, the angle between the direction the plane flies and due north.

- vp, the speed at which the plane will fly relative to the ground.
Draw a picture with a vertical line representing the desired northward flight. From the bottom of that line, draw line segment at 30.4 degrees to it, marking its length as 77.6. Finally, use compasses set to draw, from the tip of that segment, a circle with radius 203.
However, there is no "SSA" rule for congruent triangles. Unless that last segment, representing the flight of the airplane, is perpendicular to the first, there will be two points of intersection giving two solutions.

b) A man walks up a stalled escalator in time tw = 89.3 s. If he stands on the moving escalator, he reaches to top in time tm = 26.4 s. Find twm, the time it would take him to reach the top if he walked up the moving escalator?
Let the length of the escalator be L. Then his walking speed is L/89.3 and the speed of the escalator itself is L/26.4. Combining them will give him a speed of L/89.3+ L/26.4.

Homework Equations

The only one I think that is possible is the Law of Cosines.

The Attempt at a Solution

I have one vector going directly North because that is what we began with. Then there is a Northeast wind blowing, which is 30.4°. I put this vector's tail at the tail of the first vector going North. The resultant vector is then 141.8 if we use the Law of Cosines. As for the angle that the plane must fly, I'm at a loss, and have no idea. You don't necessarily have to tell me the answer, but where do I begin?

For the second question, I just have no idea at all.

I'm not too familiar with vectors, but I know that they can be used to represent forces and movement in physics. So for the escalator problem, I would approach it by breaking down the movement into vectors representing the man's walking speed and the escalator's speed. From there, I would use the Law of Cosines to find the resultant vector and then use that to find the time it would take the man to reach the top. However, I'm not sure if this is the correct approach or if there is a simpler way to solve this problem. I would appreciate any guidance or clarification on how to solve this problem.

1. How does wind affect airplane navigation?

Wind can significantly impact the speed and direction of an airplane. A headwind, which blows against the direction of travel, can slow down the plane and increase fuel consumption. A tailwind, on the other hand, can help a plane travel faster and reduce fuel usage. Crosswinds can also affect the stability of the airplane and require adjustments in the flight path.

2. How do pilots compensate for wind while flying?

Pilots use various techniques to compensate for wind while flying. They can adjust the heading of the plane to account for crosswinds, or change altitude to find more favorable winds. Pilots can also use weather reports and forecasts to plan their route and make necessary adjustments to avoid strong headwinds or take advantage of tailwinds.

3. What is the impact of wind on takeoff and landing?

Strong winds can make takeoff and landing more challenging for pilots. Crosswinds can create turbulence and affect the stability of the plane during these critical phases of flight. Pilots must be trained to handle crosswinds and make necessary adjustments to ensure a safe takeoff and landing.

4. How do escalators deal with wind in outdoor settings?

Escalators in outdoor settings are typically designed with wind deflectors to prevent strong gusts from affecting their operation. These deflectors can redirect the wind away from the escalator or create a barrier to block the wind. Some escalators also have sensors that can detect strong winds and automatically slow down or stop the escalator for safety reasons.

5. Can wind power be used to navigate planes or escalators?

Wind can be used to generate power for planes and escalators. Some planes have auxiliary power units (APUs) that use wind energy to generate electricity while in flight. Escalators in train stations or airports can also use wind turbines to generate power for their operation. However, using wind as the primary source of power for navigation is not currently feasible due to the unpredictable and variable nature of wind patterns.

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