Speed of light moving on a wall.
Consider a coordinate system where the car is at the origin, the wall is the line $x=20$, and the beam on the wall is initially at $(20,0)$ (the closest point to the car). Then the position of the beam on the wall is given by $(x(t),y(t))=(R(t) \cos(2 \pi t),R(t) \sin(2 \pi t))$. What is the appropriate choice of $R(t)$? Once you have that, you have $y(t)$ and just need to compute $y'(t)$.
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Comments
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RK01 over 1 year
While studying, I came upon this word problem: "A police car is 20 feet away from a long straight wall. Its beacon, rotating 1 revolution per second, shines a beam of light on the wall. How fast is the beam moving when it is perpendicular to the wall?" The answer key gives the answer at 40pi feet per second, but the explanation for this is not explicit. Could anyone give an explanation to reaching this answer?
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lulu over 8 yearsWhat have you tried? Can you, say, describe the location of the spot of light in terms of the angle of the beam?
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lulu over 8 yearsEasier, perhaps, to describe the y coordinate via the tangent of the angle from the origin.
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Ian over 8 years@lulu Sure. That's what winds up happening anyway, when you choose $R$. In hindsight it's more convenient to do it that way. This way was natural with the (slightly overly complicated) way that I thought of the picture. (Basically I drew a circle and a line tangent to the circle, then realized that since the line is tangent, I need a time varying radius to track the beam correctly. Then the algebra does the rest.)