# Calculating a value that depends on itself.

1. Nov 11, 2010

### tomyuey938

Hi,
I'm calculating the velocity of something that's stretching, and as it gets longer, the velocity decreases.
So to calculate the velocity at time (t):
v(t) = A / (B L(t))

A change in the distance over a small time dt will be given by:
dL=v(t) dt

So I guess the distance is:
L= integral from 0 to t of v(t) dt ?

How can I go about solving such an equation? Are itterative methods required? Can anyone give me some keywords to help my search on google? I don't really know where to start.

Thanks a lot for your time.

2. Nov 11, 2010

### tiny-tim

hi tomyuey938!
i'm confused

do you mean dL/dt = A/BL ?

if so just move the terms around: L dL = A dt/B

3. Nov 11, 2010

### tomyuey938

Hi tiny-tim,

Well, I'm trying to calculate the value of v(t) at a given value of (t). So I don't think your re-arrangement is relevant in this case, since I need v(t)=something.
But I don't know the value of L, since L is the integral of v(t) dt up to that time (t).

Does that make sense? Please do let me know if you'd like me to explain anything more, or if you think I'm mis-understanding something (which is quite possible!).

Thanks and with regards.

4. Nov 11, 2010

### maverick_starstrider

Hi tomyuey. tiny-tim's approach is indeed correct (although if you're curious the practice of solving an equation which is dependent on itself uses what's called a SELF-CONSISTENT approach). However, for this case $$\frac{dL}{dt}=v$$ so we can plug that in to get

$$v=\frac{dL}{dt} = \frac{A}{B L} \rightarrow L dL = \frac{A}{B}dt$$

Integrating one gets

$$\frac{L^2}{2}=\frac{At}{B}\rightarrow L=\sqrt{\frac{2At}{B}}$$

5. Nov 11, 2010

### tomyuey938

Hi Starstrider,

So in order to get an expression for v(t), I simply differentiate both sides by t to give:

dL/dt = SQRT(2A/B) (1/2) t^(-1/2)

Thanks also for the "self-consistent" term. This will be helpful in the future.

Thank you both so much for your help. I really do appreciate it.

Edit: but this suggests as t->0 the speed becomes infinite. Is this really correct, or have I made a mistake?

Last edited: Nov 11, 2010
6. Nov 11, 2010

### maverick_starstrider

L equals what you said not dL/dt

7. Nov 12, 2010

### Citan Uzuki

maverick_starstrider's approach is missing a constant of integration. It should be:

$$\frac{L^2}{2}=\frac{At}{B}+C \Rightarrow L = \sqrt{\frac{2At}{B} + 2C}$$

Where 2C is the square of the length at t=0. Now, if the object has zero length at t=0 (as MS implicitly assumes), then since the velocity is inversely proportional to the length, one would indeed expect the velocity to be infinite at t=0.