A thin plate glass window with a refractive index 1.5 is tapered at a 5° angle. You are looking from the outside through such a window at an angle of 30° with respect to the normal and see a person standing there behind the window (very similar to the fish tank we discussed in class). If a person is standing in the house 2m
behind the window how far of your line of sight are they really standing as opposed to where they appear to be? If at the outside position you have a gunman is looking inside under the same angle of 30 °standing trying to shoot the person in the inside of the room.How far away from the window should the person in the inside be, so that a bullet fired along the line of sight of the gunman will miss the person in the inside by 1m
To answer the first part of the question - how far the person appears to be compared to where they actually are when looking through the glass window - we can use the concept of apparent depth.
Apparent depth is given by the formula:
Apparent depth = Actual depth / Refractive index
In this case, the actual depth is 2m and the refractive index of the glass window is 1.5. Plugging in these values into the formula, we get:
Apparent depth = 2m / 1.5 = 1.33m
So, when looking through the glass window, the person appears to be standing 1.33m closer than they actually are.
For the second part of the question - how far away from the window should the person inside be, so that a bullet fired along the line of sight of the gunman will miss the person by 1m - we need to consider the refraction of light and the angle of incidence.
To make sure the bullet misses the person in the inside by 1m, we want to find the distance at which the bullet will hit the glass window. Let's call this distance "d".
Using the concept of refraction, we know that the angle of refraction is related to the angle of incidence and the refractive indices of the two media involved (in this case, air and glass). The formula for this relationship is:
sin(angle of incidence) / sin(angle of refraction) = refractive index of air / refractive index of glass
In this case, the angle of incidence is 30°, and the refractive index of glass is 1.5. Plugging in these values, we get:
sin(30°) / sin(angle of refraction) = 1 / 1.5
sin(30°) / sin(angle of refraction) = 2/3
Simplifying, we get:
sin(angle of refraction) = (3/2) * sin(30°)
sin(angle of refraction) = (3/2) * (1/2)
sin(angle of refraction) = 3/4
Using the inverse sine function to find the angle of refraction, we get:
angle of refraction = arcsin(3/4) ≈ 48.59°
Now, we can consider the path of the bullet. The bullet fired by the gunman will travel in a straight line from the outside of the window to the inside. It will enter the glass at an angle of incidence equal to the angle at which the gunman is looking (30°) and undergo refraction inside the glass.
To ensure the bullet misses the person in the inside by 1m, we need to find the distance at which the bullet will hit the glass window inside the room. Let's call this distance "d".
Using the concept of trigonometry, we can determine the value of d using the following formula:
d = 1m / tan(angle of refraction)
d = 1m / tan(48.59°)
Plugging in these values and calculating, we find:
d ≈ 0.85m
Therefore, the person in the inside of the room should be positioned approximately 0.85m away from the window to ensure that a bullet fired along the line of sight of the gunman will miss the person in the inside by 1m.