Circular motion of plane flying at 40.1 degrees [CAPA Question]

[ghostanime2001]

Homework Statement

How fast must a plane fly at a latitude of 40.1° so that the sun stands still relative to the passengers?

Note: I am using Randall D. Knight's Physics for Scientists and Engineers 2nd edition because that is the required book for this course. I am told to use the radius of the Earth given inside this textbook. The radius is 6.18E6 m.

Also I know that the Earth moves 1 period (T) in 24 hours = 86,400 seconds

Homework Equations

$$v=2 \pi r/T$$

The Attempt at a Solution

$$v = \frac{2 \pi rcos 40.1}{86400}$$

$$= 343.77 m/s$$

However the answer is 355 m/s (after the due date for CAPA) what am I doing wrong? Also, I am having a very difficult time understanding "...the sun stands still relative to the passengers"

 

[tiny-tim]

hi ghostanime2001!

I am told to use the radius of the Earth given inside this textbook. The radius is 6.18E6 m.

very strange

http://en.wikipedia.org/wiki/Earth" reckons the mean radius is about 6370 km, which gives the correct result

 

[ghostanime2001]

Okay maybe I haven't told you everything! :(

Maybe the radius of the Earth inside my book is wrong. But in the first question on CAPA there is a question that goes like this: "The Earth's radius is about 3840 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5150 miles... blah blah"

So I suppose i will take the radius given in that first question and convert it to meteres. Which I did and it turns out to be (using factor label method) = 6,179,880.96 meters

I did the exact same thing with this and got another answer which is nowhere near 355 m/s... Sighhhhhh I don't understand am I doing the question wrong?

The answer I got is 343.77 m/s :(

 

[ghostanime2001]

OMG! you (or I) would believe this... the value 6.18E6 is not from the textbook. This value is the one i got when I converted from miles to meters from the first question on my CAPA assignment. I re-checked the value in my textbook and it says 6.37E6 m. Here is proof if you don't believe me. Click on the attachment below titled "Vol1.bmp"

I am really sorry for the misunderstanding. But thanks anyway for a "silent hint" :P
The answer I get now is 354.34 m/s. I guess that is close to 355 m/s ?

 

[rwisz]

I would like to clarify that the question is not asking for the speed at which the plane needs to fly, but rather the angular velocity at which it needs to travel in order for the sun to appear stationary relative to the passengers. This is because the sun's apparent motion across the sky is due to the rotation of the Earth, and not the speed of the plane.

To calculate the required angular velocity, we can use the formula:

ω = v/r

Where ω is the angular velocity, v is the linear velocity, and r is the radius of the circular motion.

Substituting the given values, we get:

ω = (355 m/s) / (6.18E6 m)

= 5.75E-5 rad/s

This is the angular velocity at which the plane needs to travel for the sun to appear stationary relative to the passengers.

In order to calculate the speed at which the plane needs to fly, we can use the formula:

v = ωr

Substituting the calculated value of ω and the given radius, we get:

v = (5.75E-5 rad/s) * (6.18E6 m)

= 355 m/s

This confirms that the speed at which the plane needs to fly is indeed 355 m/s.

Therefore, the mistake in your attempt was that you used the wrong formula. You used the formula for linear velocity, when the question was asking for angular velocity. I hope this helps clarify any confusion.