Graphing Velocity vs Time for a Falling Rope

[knightpraetor]

SO i basically need to know what the graph of velocity vs time is..and I'm supposed ot do this numerically using maple...though if anyone has some basic intuition about what the graph should look like, even that would be nice.

Anyways, I'm unsure of how to graph it

Basically i have $$x_dot^2 = \frac {g(2bx-x^2)}{b-x}$$ and that
$$x_doubledot = g + \frac {g(2bx-x^2)}{2(b-x)^2}$$

and i thought i could just integrate the first one and get a function relating x and t, and then use that in the second equation to get an equation for velocity in terms of just time...because otherwise how am i supposed to graph it.

Anyways, for those who are wondering this graph of velocity vs time is supposed to represent a simple physical problem of a rope whose ends are attached to two ends A and B on a ceiling..and then B is cut loose and allowed to fall. So i thought the greatest acceleration would occur at the beginning and decrease as more tension occurs in the rope during the later portions with a rapid falloff in velocity at the end.

ANyways, thoughts or ideas would be nice..not really sure what to do

ps- xdot = velocity and x double dot = acceleration

 

[Mindscrape]

Huh? What is dot and double dot supposed to be, a first and second time derivative?

It looks to me like they are first and second order autonomous differential equations. What is the problem with using Maple to graph them? Maple, I haven't really used it much, is supposed to have a really good differential plotter. If that is the part you are having trouble with, you could probably use Maple's help functions and figure it out.

 

[knightpraetor]

yeah, dots are derivatives with respect to time..but which function do i integrate to get velocity? i mean i can integrate the second one (acceleration ) to get velocity, but even then, it's in terms of x rather than time..i want velocity versus time so how do i go about getting that

 

[Mindscrape]

If you solved the second differential with respect to time you would get your velocity as a function of time and x, unless that is given to you. That looks like it is pretty crappy to solve by hand, so I don't know what the solution is, but Maple should be able to do it.

Looks like you have a second order differential of the form
$$\frac{d^2 x}{dt^2}+\frac{a_1}{a_2} \frac{dx}{dt} + {a_o}x = f(x)$$
Some kind of vibration? I don't know, differentials are ugly.