Another problem from Kline

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Homework Statement


A body has acquired a downward velocity [itex] v_0 [/itex] and thereafter the acceleration is proportional to the distance traveled. Measure time and distance from the instant and place where the object has acquired the velocity [itex] v_0 [/itex] and find the formula or the distance traveled as a function of time.


Homework Equations


Suggestion in text to use

[tex] \frac{dv}{dt} = v\, \frac{dv}{dy} [/tex]


The Attempt at a Solution


I have found this problem very challenging. I'd appreciate any suggestions. I know it's long and complicated, and I am asking a lot, but if any of you have time, it really would be appreciated.

Using the suggestion in the text, I started with [itex] a = ky [/itex] which means [itex] \frac{dv}{dt} = ky [/itex] or

[tex] v\frac{dv}{dy} = ky [/tex]

My idea here was to get an expression for v, and then integrate that (as dy/dt) to get y in terms of t. But I have never really seen something like this before in the text. So perhaps I took a wrong turn here. What I tried to do was this: [tex] v\, dv = ky\, dy [/tex] and integrate this to get [tex] v^2/2 = ky^2/2 + C [/tex].

I know the book says you aren't supposed to view dv/dy (or any derivative) as a fraction, but I couldn't think of anything else to do.

On the off chance this was ok. I found [itex] C = v_0^2/2 [/itex] and so end up with [itex] v^2 = ky^2 + v_0^2 [/itex].

So (assuming this is right), I have [tex] \frac{dy}{dt} = \sqrt{ky^2 + v_0^2} [/tex] I use the theorem on reciprocals to write it as [tex] \frac{dt}{dy} = \frac{1}{\sqrt{ky^2 + v_0^2}} [/tex]

To integrate this, I set [itex] y = v_0\tan\theta/\sqrt{k} [/itex] and so [itex] dy/d\theta = v_o\sec^2\theta/\sqrt{k} [/itex] and I end up with the integral

[tex] t = \frac{1}{\sqrt{k}}\int \frac{1}{\cos\theta}\, d\theta = \frac{\log(\cos\theta)}{\sqrt{k}} = \frac{1}{\sqrt{k}}\log\left(\frac{v_0}{\sqrt{v_0^2 + y^2k}}\right) + C [/tex]

So

[tex] De^{\sqrt{k}t} = \frac{v_0}{\sqrt{v_0^2+y^2k}} [/tex]

and

[itex] D = 1/v_0 [/itex].

Here is where I stopped, since it is obvious that this is not going to lead to the correct answer as given in the book, which is

[tex] y = \frac{v_0}{2\sqrt{k}}(e^{\sqrt{k}t} - e^{-\sqrt{k}t}) [/tex]

Thanks again, if anyone has the fortitude to slog through all this.

Sheldon
 
Last edited:

Answers and Replies

  • #2
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Now I see a mistake in my integration of 1/cos. Let me think on this some more, and I will post again. Sorry to bother everyone.

Sheldon
 
Last edited:
  • #3
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Yes, that was the problem. Well, I managed on my own for once. Sorry to bother anyone who read this.

Thanks
Sheldon
 

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