C

#### caffeine

##### Guest

Someone asked me a basic physics question, and I'm not believing my answer, even though it appears to have the correct limiting behavior. This is driving me completely insane.

Suppose I stand on a sidewalk, 10 feet from the middle of an infinitely long and straight road. I have a flashlight in my hand, and I rotate the flashlight at some constant angular speed [tex]\omega[/tex]. The question posed to me was "what is the speed of the light's image along the middle of the road, 30 feet from where I'm standing"?

I'm standing at point O with my flashlight. I place an origin on myself with y pointing "up" and x pointing to the right. So [tex]R=OA=10[/tex]ft and [tex]R'=OB=30[/tex]ft. The velocity of the lightbeam, as a function of position should be:

[tex]

\vec{v}(\vec{r}) = \omega\,r \left[ \hat{x}\,\cos(\theta) - \hat{y}\, \sin(\theta) \right]

[/tex]

Let [tex]\theta[/tex] be measured from the vertical. At point B,

[tex]

\theta = \cos^{-1}\left(\frac{R}{R'}\right)

[/tex]

And I'm already in trouble. I want the speed of the flashlight along the middle of the road, so I'm asked for [tex]\vec{v}(B)\cdot\hat{x}[/tex]. If you plug in my [tex]\theta[/tex] into my expression for the flashlight's velocity and take the dot product, you get something constant:

[tex]

\vec{v}(B)\cdot\hat{x} = \omega \, R

[/tex]

However, I refuse to believe this. This is saying the flashlight's x component of velocity is constant. My physical intuition says this is nonsense. I can certainly integrate the time it takes for the flashlight to go the entire (infinite) distance of the road. The time *should* be the time it take for me to rotate through [tex]\pi[/tex] radians. That time should be [tex]\pi / \omega[/tex], however, if the x component of velocity is constant, that time will be infinite.

The full answer (what is the velocity of the flashlight anywhere on the middle of the road) is:

[tex]

\vec{v}(R') = \omega \left[\hat{x}\,R - \hat{y} \sqrt{R'^2 - R^2} \right]

[/tex]

which of course becomes [tex]-\infty[/tex] when [tex]R'\rightarrow\infty[/tex] and [tex]\infty[/tex] when [tex]R'\rightarrow-\infty[/tex]. That's expected. The speed of this result is [tex]|\vec{v}(R')| = \omega \, R'[/tex], which is correct also. But how is it possible that the x component remains the same whether [tex]R'\rightarrow\infty[/tex] or [tex]R'\rightarrow R[/tex]. Does that make sense?

I can't see where I went wrong. What's going on here?

Suppose I stand on a sidewalk, 10 feet from the middle of an infinitely long and straight road. I have a flashlight in my hand, and I rotate the flashlight at some constant angular speed [tex]\omega[/tex]. The question posed to me was "what is the speed of the light's image along the middle of the road, 30 feet from where I'm standing"?

Code:

```
A B road
* ------------- * ---------------
.
.
*O
```

[tex]

\vec{v}(\vec{r}) = \omega\,r \left[ \hat{x}\,\cos(\theta) - \hat{y}\, \sin(\theta) \right]

[/tex]

Let [tex]\theta[/tex] be measured from the vertical. At point B,

[tex]

\theta = \cos^{-1}\left(\frac{R}{R'}\right)

[/tex]

And I'm already in trouble. I want the speed of the flashlight along the middle of the road, so I'm asked for [tex]\vec{v}(B)\cdot\hat{x}[/tex]. If you plug in my [tex]\theta[/tex] into my expression for the flashlight's velocity and take the dot product, you get something constant:

[tex]

\vec{v}(B)\cdot\hat{x} = \omega \, R

[/tex]

However, I refuse to believe this. This is saying the flashlight's x component of velocity is constant. My physical intuition says this is nonsense. I can certainly integrate the time it takes for the flashlight to go the entire (infinite) distance of the road. The time *should* be the time it take for me to rotate through [tex]\pi[/tex] radians. That time should be [tex]\pi / \omega[/tex], however, if the x component of velocity is constant, that time will be infinite.

The full answer (what is the velocity of the flashlight anywhere on the middle of the road) is:

[tex]

\vec{v}(R') = \omega \left[\hat{x}\,R - \hat{y} \sqrt{R'^2 - R^2} \right]

[/tex]

which of course becomes [tex]-\infty[/tex] when [tex]R'\rightarrow\infty[/tex] and [tex]\infty[/tex] when [tex]R'\rightarrow-\infty[/tex]. That's expected. The speed of this result is [tex]|\vec{v}(R')| = \omega \, R'[/tex], which is correct also. But how is it possible that the x component remains the same whether [tex]R'\rightarrow\infty[/tex] or [tex]R'\rightarrow R[/tex]. Does that make sense?

I can't see where I went wrong. What's going on here?

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