I had to babysit tonite, so I got bored. So here is my problem statement:
A length L of perfectly flexible rope is coiled at the edge of a cliff of height H (H<L). One end of the rope begins to fall off the edge under gravity, with the rest of the rope smoothly uncoiling as each coil is pulled down in turn. Determine the speed of the rope as it falls from the time that the leading end of the rope hits the ground to the time that the trailing end of the rope leaves the cliff. Neglect friction.
First, I parameterized the system with a parameter y which represents the amount of rope that has already hit the ground. The parameter y ranges from 0 to L-H. I then wrote the conservation of energy expression in terms of y:
g H L \rho =\frac{1}{2} H \rho v(y)^2+g H \left(-\frac{H}{2}+L-y\right)<br />
\rho +\frac{1}{2} \rho \int_0^y v(\psi )^2 \, d\psi
Where \rho is the linear density of the rope, and v is the velocity of the rope as a function of y. The left hand side is the initial potential energy before the rope begins to fall. The first term on the right hand side is the kinetic energy in the falling rope, the second term is the gravitational potential energy, the third term is the (plastic) collision energy as the rope hits the ground.
Setting y=0 and solving for v(0) we obtain:
v(0)=\sqrt{g H}
Differentiating both sides wrt y we obtain the following differential equation:
0=\frac{1}{2} \rho v(y)^2+H \rho v'(y) v(y)-g H \rho
Solving this differential equation with the above initial condition we obtain:
v(y)=\sqrt{2 g H-e^{-\frac{y}{H}} g H}
Thus there is no constant velocity although the velocity exponentially approaches:
v(\infty )=\sqrt{2} \sqrt{g H}=\sqrt{2} v(0)
Bottom line, there is no problem with conservation of energy here, only a problem with misapplying a formula and making several assumptions about the results without doing the math. Conservation of energy has been proved in general, as mentioned by arildno and others, but it is not terribly difficult to come up with a complicated scenario with so many variables and details that even the best "problem solver" might work it incorrectly. Whenever you find that some complicated scenario seems to violate some known physical law you can be sure that "the devil is in the details". There is always some ambiguity in the problem set up or some error in the work or assumptions. If you really want to demonstrate a violation of a physical law look for a very simple and clear example where the violation cannot possibly be due to any other factor.
) And I think you'll get more useful--and more interesting--insight by trying to figure out the energy loss, rather than just by assuming energy conservation.
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