1. ## Aeroplane

Hello all, I am a British Maths GCSE student working towards the final exam and one day this question cropped up in a mock Maths exam I have done recently. I could not work it out and think it might have cost a percentage of the mark, and was wondering whether it would crop up again.

I try to represent it here and wonder whether any of you guys could help with offering a solution as I could not for revision purposes.
Any help concerning the answer would be greatly appreciated.
Here goes.

The Earth’s circumference around the equator is 24907.55 miles
(1 mile = 1.609344 km)

A plane flies 11km above the surface of the Earth and flies westward following the line of the equator. If the Plane is travelling at 510km per hour, how long will it take for the plane to reach the same place above the Earth’s surface from where it started from? (i.e. at the beginning of the question)

I think you have to consider the rotation of the Earth and km vs miles.
And I think the solution lies in finding the radius of the Earth and adding 11km to it, but I'm not sure.

When it says "the same place above the surface", I believe it is talking about the same place over the Earth's surface at the beginning and at the end as opposed to a complete circumnavigation of the Earth.
i.e. Over Australia at the beginning and at the end.
Any help with this would be greatly appreciated.

2. Originally Posted by Neod4000
Hello all, I am a British Maths GCSE student working towards the final exam and one day this question cropped up in a mock Maths exam I have done recently. I could not work it out and think it might have cost a percentage of the mark, and was wondering whether it would crop up again.

I try to represent it here and wonder whether any of you guys could help with offering a solution as I could not for revision purposes.
Any help concerning the answer would be greatly appreciated.
Here goes.

The Earth’s circumference around the equator is 24907.55 miles
(1 mile = 1.609344 km)

A plane flies 11km above the surface of the Earth and flies westward following the line of the equator. If the Plane is travelling at 510km per hour, how long will it take for the plane to reach the same place above the Earth’s surface from where it started from? (i.e. at the beginning of the question)

I think you have to consider the rotation of the Earth and km vs miles.
And I think the solution lies in finding the radius of the Earth and adding 11km to it, but I'm not sure.

When it says "the same place above the surface", I believe it is talking about the same place over the Earth's surface at the beginning and at the end as opposed to a complete circumnavigation of the Earth.
i.e. Over Australia at the beginning and at the end.
Any help with this would be greatly appreciated.
I would imagine that if the plane is traveling at a speed of 510 km/h then this speed would be measured in reference to a point on the Earth, which is rotating. Thus this speed already takes into account the rotation of the Earth.

As far as the circle the plane flies in, yes, we need to add the height of the plane above the surface. So the circumference of such a circle will be
$C = 2 \pi (R_E + h)$
where $R_E$ is the Earth's radius and h is the height of the plane above the Earth.

-Dan

3. Hello,
Originally Posted by Neod4000
...

The Earth’s circumference around the equator is 24907.55 miles
(1 mile = 1.609344 km)
then the circumference at an altitude of 0 km is:

$c_0=24907.55\ mi \cdot 1.609344\ \frac{km}{mi}= 40084.81615\ km$

Originally Posted by Neod4000
A plane flies 11km above the surface of the Earth and flies westward following the line of the equator.
If R is the radius of the earth then the circumference at an altitude of 11 km is:

$c_{11}=2 \cdot \pi (R+11)=\underbrace{2 \pi R}_{\text{that's }c_0}+22 \pi$

That means the plane has to cover a greater distance.
The traveling time would be 78 h 44 min

Originally Posted by Neod4000

I think you have to consider (A) the rotation of the Earth and km vs miles.
And I think the solution lies in finding the radius of the Earth and adding 11km to it, but I'm not sure.

i.e. (B) Over Australia at the beginning and at the end.
to (A): You only have to consider the rotation when you are going north (or south) because the speed of a point of the surface depends on the length of the radius of the actual latitude.

to (B): That would be quite difficult to push Australia under the equator

4. Thank you all very very much, you are very helpful people.
Originally the figure I got was 0.45692753 seconds, which is slightly out of perspective.

5. Originally Posted by earboth
(A): You only have to consider the rotation when you are going north (or south) because the speed of a point of the surface depends on the length of the radius of the actual latitude.
Good thought. I hadn't considered that case.

-Dan

6. Wait, sorry, 78 hrs can't be right.
The plane is travelling westwards (or clockwise) and the Earth is traveling anticlockwise because the Sun rises in the east, so logically, the Earth should be doing half the work for the plane anyway.

You've said yourself that
Originally Posted by earboth
You only have to consider the rotation when you are going north (or south) because the speed of a point of the surface depends on the length of the radius of the actual latitude.
But when a point is moving towards you on the same latitude as you are on that means that you should meet each other at some point without having to do a complete circumnavigation of the Earth.

7. Originally Posted by Neod4000
Wait, sorry, 78 hrs can't be right.
The plane is travelling westwards (or clockwise) and the Earth is traveling anticlockwise because the Sun rises in the east, so logically, the Earth should be doing half the work for the plane anyway.

You've said yourself that

But when a point is moving towards you on the same latitude as you are on that means that you should meet each other at some point without having to do a complete circumnavigation of the Earth.
Hello,

if your statement were true you only have to jump vertically into the air and wait until New York is passing under your feet ...

The plane on the runway has the same movement as the earth that means it is moving to the east. What you described is the movement which an observer in outer space will see. If the plane has actually the speed zero it will vanish with a supersonic boom out of sight. A point of the equator is moving with a speed of 1667 km/h and that's more than the speed of noise. If you are for instance in Singapore you never will notice your speed because you are moving with the same speed as all the houses, men, etc. around you. The relative speed is zero and that's the only speed you can observe.

8. But if the plane is moving one way and the Earth is moving another then surely that will have an effect on the actual outcome if the Earth is doing a portion of the plane's work for it.

9. Originally Posted by Neod4000
But if the plane is moving one way and the Earth is moving another then surely that will have an effect on the actual outcome if the Earth is doing a portion of the plane's work for it.
Is the plane moving faster in one direction than the other? Sure it is, it has to be relative to an observer that isn't rotating along with the Earth. But the question deals with a plane traveling at a fixed speed which is measured with respect to the Earth as a stationary surface. So it will take both planes the same time to go all the way around.

-Dan

10. But what if the question did consider the rotation of the Earth?
What would happen then?

11. Originally Posted by Neod4000
But what if the question did consider the rotation of the Earth?
What would happen then?
If both planes are measured to travel at the same speed in regard to a fixed point not rotating with the Earth, then the plane that flies against the Earth's rotation will get there first.

-Dan

12. There's only one plane in the question.
All I mean is whether or not it would take 78 hours 44 minutes for a plane to reach the same spot over the Earth as it was when it started off if the Earth was revolving in the opposite direction that the plane was going,
i.e.

So the Earth would be doing some of the work for the plane.
Would this be accurate?

13. Originally Posted by Neod4000
There's only one plane in the question.
All I mean is whether or not it would take 78 hours 44 minutes for a plane to reach the same spot over the Earth as it was when it started off if the Earth was revolving in the opposite direction that the plane was going,
i.e.

So the Earth would be doing some of the work for the plane.
Would this be accurate?
And what I am saying is that if you measure the speed of the plane referenced to a point that is not rotating with the Earth that a plane flying with the rotation of the Earth will take longer than a plane flying against the rotation of the Earth. Yes, you are essentially correct.

-Dan

14. So am I correct in thinking that in that situation the answer would not be 78 hours 44 minutes?

15. Originally Posted by Neod4000
So am I correct in thinking that in that situation the answer would not be 78 hours 44 minutes?
I haven't bothered to calculate the time so I don't know off hand if it is correct or not. Since the plane is going to be measuring its speed relative to the ground, it does not matter which way it travels: it will take the same amount of time to go around once.

-Dan