Rainbow Visible to Hiker on Isolated Mountain Peak: 0.49 Fraction

[Abhishek.S]

A hiker stands on an isolated mountain peak near sunset and observes a rainbow formed by water droplets in the air 8 km. away. The valley is 2 km. below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker?

I tried to solve the problem with the fact that the deviations of red light and violet light are 42 and 40 degrees resp.
I got the width of the rainbow he sees as 8tan42-8tan40 = 0.49
But what is the total width?

 

[andrevdh]

I just want to mention first that stating that the rainbow forms at a distance of 8 km is a bit artificial, since it really forms at infinity (Well actually there is nothing out there, the rainbow forms on your retina, so each observer has his own personal rainbow inside of his eye! But if you were to stand next to me and I say "Do you see the rainbow there?" you would agree with me. Another observer flying over us in his airoplane would actually see the rainbow as a circle in another place. You can even photograph it and it will appear on the film, so we all say seeing is believing, but it still isn't actually out there! Some people might disagree with this statement of mine, this is my personal view on the subject. It makes you think doesn't it?), but that aside with the given info one can calculate the radius of the red circle of the rainbow at this distance, $r_r$. That enables you to evaluate the angle $\alpha$ that is cut off by the horizon from this circle and finally the arc that is below the horizon $s_{below}$.

There probably is a much more elegant way of calculating this, but for now it eludes me.

 

[kyle8921]

I would first like to clarify the assumption made in this problem. The problem states that the hiker is observing a rainbow formed by water droplets in the air. However, it does not specify the exact conditions and environment that would allow for such a rainbow to form. I would need more information about the weather conditions, humidity levels, and other factors in order to accurately calculate the visibility of the rainbow.

That being said, let's assume that the conditions are ideal for a rainbow to form and the visible rainbow is a complete circular arc. In this case, the total width of the rainbow would be 360 degrees. However, since the hiker is located on an isolated mountain peak and the valley is 2 km below, the hiker's perspective would only allow for a portion of the rainbow to be visible.

Using the information provided, we can calculate the distance between the hiker and the rainbow as 8 km. This means the hiker is at the center of a circle with a radius of 8 km, and the rainbow would form a portion of the circumference of this circle. Using basic geometry, we can calculate the arc length visible to the hiker as 8(42-40) = 0.49 km.

In order to calculate the fraction of the complete circular arc visible to the hiker, we can divide the arc length visible (0.49 km) by the total arc length (360 degrees). This gives us a fraction of 0.0014, which is approximately 0.14%. This means that the hiker would only be able to see a very small portion of the complete circular arc of the rainbow.

In conclusion, the fraction of the complete circular arc of the rainbow visible to the hiker on the isolated mountain peak is approximately 0.14%. However, this calculation is based on certain assumptions and may vary depending on the actual conditions of the environment. I would recommend further research and data collection to accurately determine the visibility of the rainbow in this scenario.