What Is the Oscillation Period for a Particle in a Hyperbolic Potential Field?

[renegade05]

Homework Statement

Determine the oscillation period, as a function of energy E, when a particle of mass m moves in a field with potential energy $V(x)=-V_0/cosh^2(\alpha x)$, for $-V_0 < E < 0$ with Vo positive.

(a) First show that, with s=sinh (α x) and determining the appropriate smax ,the period satisfies

$T=\frac{2\sqrt{2m}}{\alpha\sqrt{E}}\int_{0}^{S_{max}}\frac{ds}{\sqrt{s^2}+\frac{E+U_o}{E}}$

Homework Equations

Not sure where to start. My text-book is of no help when it comes to this question

The Attempt at a Solution

I am not sure where to start...

Can someone help me with this one. thanks!

 

[TSny]

Show some attempt. You know you need to use energy ideas. What can you say about the relation between kinetic energy, potential energy, and total energy E?

 

[renegade05]

T + v = e... Don't see what the would do?

 

[TSny]

Suppose you could express the speed v as a function of x. Note v = dx/dt. So, you would have dx/dt = some function of x.

 

[renegade05]

Ya I get that, i just don't know how to get smax

 

[TSny]

What can you say about the value of the potential energy V(x) when s = s_max?