How is the integral expression for contact time derived?

[rohanc]

Don't know if this is the correct place to post this, this is not an assignment question, but I am terribly stuck with this.

While going through the derivation of contact time for a hertzian contact as given in problem 3 at the following link http://s17.postimg.org/t1kq6mlxr/Capture.png , I am not able to understand how the integral form for contact time has come into picture. I understand that the twice the integral of displacement over velocity from x=0 to x=x0 gives the total contact time. But can anyone please explain the in-between steps to get the same (how did the 1/sqrt(1-(x/x0)^5/2) come about? I understand that this is a very trivial problem but it will be a great help to understand the steps.

 

[Ray Vickson]

Don't know if this is the correct place to post this, this is not an assignment question, but I am terribly stuck with this.

While going through the derivation of contact time for a hertzian contact as given in problem 3 at the following link http://s17.postimg.org/t1kq6mlxr/Capture.png , I am not able to understand how the integral form for contact time has come into picture. I understand that the twice the integral of displacement over velocity from x=0 to x=x0 gives the total contact time. But can anyone please explain the in-between steps to get the same (how did the 1/sqrt(1-(x/x0)^5/2) come about? I understand that this is a very trivial problem but it will be a great help to understand the steps.

Don't know if this is the correct place to post this, this is not an assignment question, but I am terribly stuck with this.

While going through the derivation of contact time for a hertzian contact as given in problem 3 at the following link http://s17.postimg.org/t1kq6mlxr/Capture.png , I am not able to understand how the integral form for contact time has come into picture. I understand that the twice the integral of displacement over velocity from x=0 to x=x0 gives the total contact time. But can anyone please explain the in-between steps to get the same (how did the 1/sqrt(1-(x/x0)^5/2) come about? I understand that this is a very trivial problem but it will be a great help to understand the steps.

You have an equation of the form
$$\left( \frac{dx}{dt}\right)^2 + c\, x^{5/2} = K$$
so
$$\frac{dx}{dt} = \sqrt{K -c\, x^{5/2}}$$
(assuming $dx >0$), hence
$$dt = \frac{dx}{\sqrt{K -c\, x^{5/2}}}$$