Calculating the Speed of a Plane for a Stationary Sun at 26.3° Latitude

[Tina20]

Homework Statement

How fast must a plane fly at a latitude of 26.3° so that the sun stands still relative to the passengers?

Homework Equations

I have no idea how I can solve this question. I would like to draw a free-body diagram, but I don't know how to do that either. If I could at least draw a diagram, it would help me to think through this question.

The Attempt at a Solution

I believe that the plane is 26.3 degrees above the equator, flying around the earth. So it is a circular motion question.

I don't have an acceleration, now do I have a period, but I think that the period might be one rotation around the earth, so 24 hours is one period? Also, the radius of the Earth is 6.37e6 m. angular velocity = speed/radius and angular velocity = 2pi/period

 

[willem2]

You indeed have a period of 24 hours.

The radius of the circle at a latitude of 26.3 degress is the distance from the Earth's axis to any point of this circle.
draw a picture of a crosssection through the Earth along the Earth's axis.

 

[Tina20]

Thank you! I drew the free body diagram, found the radius using 6.37x10^6 cos 26.3 deg. and then substituted the radius and the period (T) --> (24hours = 86400 sec) into v= 2pir/T and got the answer :)

 

[ghostanime2001]

Why do u need to know the freebody diagram?

 

[rwisz]

As a scientist, it is important to approach any problem systematically and logically. In this case, we are trying to calculate the speed of a plane at a specific latitude so that the sun appears to be stationary to the passengers.

First, we need to understand the motion of the plane and the sun. The plane is flying in a circular path around the Earth, while the sun appears to move across the sky due to the Earth's rotation. At the latitude of 26.3°, the plane is still moving in a circular path, but the angle at which it intersects the Earth's axis is different compared to the equator.

To solve this problem, we can use the equation v = ωr, where v is the speed of the plane, ω is the angular velocity, and r is the radius of the circular path. We know that the angular velocity is equal to 2π divided by the period of rotation, and the period of rotation is 24 hours. We also know that the radius of the Earth is 6.37e6 m.

Therefore, we can calculate the speed of the plane by plugging in these values into the equation v = (2π/24 hours) x (6.37e6 m). This gives us a speed of approximately 463.8 m/s. This means that the plane must fly at a speed of 463.8 m/s in order for the sun to appear stationary to the passengers at a latitude of 26.3°.

In terms of drawing a free-body diagram, it may not be necessary in this case as we are dealing with circular motion and not forces. However, if you were to draw one, you could show the forces acting on the plane, such as the gravitational force from the Earth and the centripetal force keeping the plane in its circular path.

Overall, this problem can be solved by understanding the motion of the plane and the sun, and using the appropriate equations to calculate the speed of the plane. It is important to approach any problem in a systematic and logical manner, and to use all the given information to arrive at a solution.