Can we solve for x(t) and y(t) with respect to time?

[Sotiris Michos]

Homework Statement

I would like to ask a question I was discussing the other day with a friend of mine. Suppose you have a point mass m, sliding on the friction-free curve $y = e^{-x}$ starting from position $x(0) = 0$ and $y(0) = 1$ with zero initial velocity. Can we find in closed-form the exact position $(x,y)$ with respect to time?

2. The attempt at a solution
Going down the usual path, from conservation of energy I found that $v = \sqrt{2g(1-y)}$. Assuming $φ$ is the angle between $\vec{v}$ and the downward vertical direction (the direction of negative ordinates), and using that $\dot{x} = v \sin(φ)$, $\dot{y} = v \cos(φ)$, $φ = \tan^{-1}(e^{x})$, I reached

$$\dot{x} = \sqrt{2g}\sqrt{\frac{e^{2x}-e^{x}}{e^{2x}+1}}$$
$$\dot{y} = -y\sqrt{2g}\sqrt{\frac{1-y}{1+y^2}}$$

From these two, I need to solve only one. Choosing the first, separating $dx$ and $dy$ and integrating, yields

$$\int_0^{x(t)}\sqrt{\frac{e^{2x} + 1}{e^{2x} - e^{x} }}dx = \sqrt{2g}t$$

However, plugging the integral in the LHS in Mathematica doesn't compute anything, most likely meaning that this integral is not solvable in closed form, let alone invertible in closed-form.

Choosing now the second equation and doing the same yields:
$$\int_1^{y(t)}\sqrt{\frac{1+y^2}{1 - y}}\frac{dy}{y} = - \sqrt{2g}t$$

Again, trying to compute the integral in the LHS, gives an assortment of complex elliptic integrals that are gain not invertible in some closed-form. Any thoughts about what can be done?

 

[BvU]

You reached $$
\dot{x} = \sqrt{2g}\sqrt{\frac{e^{2x}-1}{e^{2x}+1}}$$
but I reach something else ( from $\ \ e^{2x}(1-e^{-x})\ \$ under the square root in the numerator ).

 

[Sotiris Michos]

You reached $$
\dot{x} = \sqrt{2g}\sqrt{\frac{e^{2x}-1}{e^{2x}+1}}$$
but I reach something else ( from $\ \ e^{2x}(1-e^{-x})\ \$ under the square root in the numerator ).

Thank you for replying BvU! You are absolutely correct $\dot{x}$ is not the one I wrote but
$$\dot{x} = \sqrt{2g}\sqrt{\frac{e^{2x}-e^{x}}{e^{2x}+1}} = \sqrt{2g}\sqrt{\frac{1-e^{-x}}{1+e^{-2x}}}$$
I also noted that $\dot{y}$ needs a minus in order for $\dot{y} = -e^{-x}\dot{x} = -y \dot{x}$ to hold. I will correct the initial post! However, after this correction, things get even more challenging, since the integral now is not analytically computable!

 

[BvU]

Did someone promise you there would be an elegant looking analytical solution ?
If you want that, you might look at a cycloid instead of an exponential

 

[Sotiris Michos]

Did someone promise you there would be an elegant looking analytical solution ?
If you want that, you might look at a cycloid instead of an exponential

Haha! None indeed BvU! As I said, it is just an inquiry I had with a friend of mine the other day. It surprised me that the only method I could find involved integration and then inversion of the result so I thought that I might be going down the wrong path. At best, there should be a way to acquire the solution in some integral form and then let numerical integration take over!

Thank you for your warm welcoming message!