Physics

[h3ll0]

a plane flying due north at 100 m/s is blown by a 50 m/s strong wind due east. what is the plane's resultant velocity?

[Kidas]

Hey Hymie.

That's always a cool way to deal with teachers - I'm guessing it didn't make you popular, though.

I trust I managed to answer the question to your satisfaction.

[Hymie]

My former teachers would have responded to my above observation ‘When did you last see an aircraft fitted with a large sail?’

The answer to the question is anywhere between due north (where the easterly wind is having no effect (force) on the aircraft) and almost due east (this would occur where the engine thrust to achieve 100m/s forward travel is minimal compared to the force of the easterly wind on the airframe.

[jim360]

That's cool and all, but the question clearly isn't about anything other than testing vector addition skills.

[Kidas]

Hi Hymie

I see what you mean about having teachers who were not of the right calibre.
They have clearly failed you.

I can see, rather like Spath, that you have developed a common-sense, Person-in-the-Street view of how these things can be approached.

That’s a good thing. It’s been over 30 years since I learned how to add vectors together. I’ve used the technique only once to solve a real-world problem, even though I’m a professionally-trained engineer, who understands how, where and why to apply that technique.

An ordinary Person-in-the-Street would rarely have any use for it. That means your common-sense approach to the situation works perfectly well for you and for everyone else who has managed to evade learning the formal process of adding vectors together using geometry.

There is only one situation where that common-sense approach breaks down. We have that situation here. We are dealing with a stylised, impractical mathematical exercise, specifically designed to test the student’s understanding of the formal process of adding a couple of vectors together using geometry.

I say stylised: we have an aircraft flying at 100 m/s. That’s about 225mph. And we have a crosswind of 50 m/s. That’s about 110 mph. This is not a real situation. Jet aircraft tend to fly a lot faster, except on take-off and landing. No pilot of a prop-driven aircraft would choose to fly in those conditions.

This is what maths tutors do: invent highly stylised problems that phrase a mathematical problem in every-day language, seeking to pretend that this could be a real-life situation that the student has to solve. It’s not a real-world situation. It’s a stylised problem intended to give the student practice at solving similar stylised problems.

I gave the conventional solution to the problem. Then you introduced the idea of transients. That’s a fun diversion from typical vector addition problems.

You also introduced the idea of a sail on the aircraft. That’s not a ludicrous fantasy, it’s the reality. The large tailfin on most commercial airliners is, essentially, a sail. It looks like one and it behaves like one, in that it gives aerodynamic stability.

Since you introduced the idea of a sail, perhaps you understand sailing. I ask you to imagine a sailboat – say a Fireball or a 505 – with the wind directly behind it, on a calm reservoir with no engines. Can that boat travel faster than the wind? Even with a huge spinnaker and no centreboard, can that boat out-run the wind? (remember, this is with the wind directly behind)

I put it to you that there is no way that the boat can out-run the wind, using only wind power (and on a run).

The same situation applies to the aircraft being blown East-West by the wind. That tailfin is catching the cross-wind like a big spinnaker sail.

In the steady-state, the wind is blowing the aircraft in the East-West direction like that sailboat. But the wind cannot push the aircraft faster than the 50 m/s wind in the East-West direction.

The engines are driving it forwards in the North-South direction at 100 m/s

In one second, how far does the plane travel forward? 100m
In the same second, what is the greatest distance it can travel in the East-West direction? 50m

You can draw triangles using those numbers to find out the theoretical speed and direction of travel resulting from the combined effect of those two movements.

That’s the basis of solving these rather unrealistic, stylised problems.

It’s OK that you take a common-sense approach. You’re never going to need this kind of mathematical technique.

It’s just that asserting your common-sense belief as a response to a formalised maths problem may confuse the poor OP who is just trying to grasp the fundamentals of vector addition.

In the world of stylised mathematical problems, I am sorry to say it, but your answer is wrong.

If, like your teachers, you find me of insufficient calibre to explain it properly; I apologise. Perhaps others can help.