Here is the curve y = 1 - Cos[x] shown with several vertical light beams: Clear[
ID: 1622689 • Letter: H
Question
Here is the curve y = 1 - Cos[x] shown with several vertical light beams:
Clear[f, x, t];
f[x_] = 1 - Cos[x];
x[t_] = t;
y[t_] = f[t];
curve = ParametricPlot[{x[t], y[t]}, {t, -1, 2}, PlotStyle -> {{Thickness[0.01], Blue}}, AxesLabel -> {"x", "f[x]"}];
beams = Table[Vector[{x[t], y[t]} - {x[t], 3}, Tail -> {x[t], 3}, VectorColor -> Red], {t, -1, 2, 3/8}];
setup = Show[curve, beams, PlotRange -> All]
1.Throw the plots of the reflected light into the plot, and use your plot to study the question:
Which parts of the curve plotted above are good at concentrating the reflected light? Which parts are not so good?
2.Stealth bombers and fighters were designed to try to resist detection by radar. When they were first unveiled, lots of folks asked why the skin of the planes is made with flat panels and absolutely no curved indentations. Look at your answer to part a), turn on your brain, and speculate about why stealth bombers and fighters are designed this way.
This is the figure
Pls answer both parts
outM 1.0 f[x] 3.0 2.5 2.0 1.5 l .0 0.5 0.5 1.0 1.5 2.0Explanation / Answer
This is a very nice question for coordinate geometry. It is important to understand the concepts behind light rays to solve the question. The concepts of light and its reflection properties is needed in this.
Reflection of light: Any light ray incident upon a reflecting surface follows a few laws. The angle between light ray and the normal line (line perpendicular to the surface) is called angle of incidence. Similarly the angle between the normal line and the reflected ray is called angle or reflection.
"For any reflecting surface, angle of incidence is equal to angle of reflection"
Now in the above problem, we know the incident rays are parallel to the y axis. Hence their slope is also known = infinity.
The equation of the reflecting surface is known. We need to develope an equation for the slope of normal line w.r.t x.
Now, at the given surface, the slope for any value x is given by dy/dx = d(1-cosx)/dx = sinx
To find the slope of normal line, we need to use the following theorem:
Product of slopes of perpendiclar lines = -1
hence, slope of normal line is given by slope_normal = -1/sinx
Now here after, we just need to use another mathematical concept of finding the reflection of a line w.r.t. another line. This is something you must have been taught in mathematics class for coordinate geometry and vectors. Please refer a standard book for coordinate geometry for this. I could help explaining that but its a large subject within itself and out of scope of physics. Please revert back if you need help in that.
Finally when you see the results of the above activity and plot the reflected lines, you will see that the parts of the reflecting surface which are curved i.e. nearby y axis, concentrate light more than the parts which are relatively straight.
b) The above observation helps us to understand that the stealth bombers do not have curved surfaces so as to prevent concentration of the reflected lradar rays. If the reflected radar rays concentrate at a single point, the radar receiver will know that there is an object at that spot. Whereas if the reflected radar rays disperse in various directions, the concentration of rays wont happen and hence the radar receiver will not be able to detect the plane.