Between this and the next assignment, we want to get a better under- standing of how light interacts with the eye. Here are two questions to get us started, focused on diffraction (i. e., the spreading of light when it passes through a narrow opening). A. To regulate the intensity of light reaching our retinas, our pupils1 change diameter anywhere from 2 mm in bright light to 8 mm in dim light. Find the angular resolution of the eye for 550 nm wavelength light at those extremes. In which light can you see more sharply, dim or bright

θ₁ = 3.35 10⁻⁴ rad ,  θ₂ = 8.39 10⁻⁵ rad

Explanation:

This is a diffraction problem for a slit that is described by the expression

       sin θ = m λ

the resolution is obtained from the angle between the central maximum and the first minimum corresponding to m = 1

      sin θ = λ / a

as in these experiments the angle is very small we can approximate the sine to its angle

        θ = λ / a

In this case, the circular openings are explicit, so the system must be solved in polar coordinates, which introduces a numerical constant.

       θ = 1.22 λ / D

where D is the diameter of the opening

 let's apply this expression to our case

indicates that the wavelength is λ = 550 nm = 550 10⁻⁹ m

the case of a lot of light D = 2 mm = 2 10⁻³ m

       θ₁ = 1.22 550 10-9 / 2 10⁻³

       θ₁ = 3.35 10⁻⁴ rad

For the low light case D = 8 mm = 8 10⁻³

      θ₂ = 1.22 550 10-9 / 8 10⁻³

      θ₂ = 8.39 10⁻⁵ rad

i don't understand is there any other way you can rephrase it plz let me know in the comments and i will do it.

explanation:

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.