Newton's law of resistance Anyone can give me some clues on how this can be proved? What kind of drag force is used? 

A particle is projected vertically upward with an initial speed $v_0$ near the Earth's surface. Show that if Newton's law of resistance applies, then the speed of the particle when it returns to it initial position is
  $$v_0v_{\text{term}}\sqrt{v_0^2+v_{\text{term}}^2}$$
  where $v_{\text{term}}$ is the terminal speed.

 A: Apparently "Newton's law of resistance" refers to quadratic drag. 
Drag isn't energy conserving. It constitutes a non-holonomic system ("a system whose state depends on the path taken to achieve it"). Unfortunately this means that there isn't a large range of easy routes of conservation theorems or clever tricks. You have to solve the whole dynamics problem! That's why this ends up being much harder than normal mechanics problems. So, the process is:


*

*Rewrite the ODEs $m\ddot{x}=-\beta \dot{x}^2-m g$ (object moving upwards) and $m\ddot{x}=\beta \dot{x}^2-m g$ (object falling downwards) in terms of the terminal velocity instead of $\beta$ (for which $\beta v_t^2-m g=0$)

*Solve both ODEs (divide through by the right-hand term and integrate with respect to time).

*Find the maximum height acheived from the upwards-moving ODE

*Plug that in to the initial height on the downwards-moving ODE

*Solve for the time that the object hits the ground and use that to solve for the velocity when the object hits the ground.


The whole thing simplifies to $$v_f=-\frac{v_t v_0}{\sqrt{v_0^2+v_t^2}}$$
If you do your ODE-solving with some foresight you might be able to skip solving for the time and then plugging that back in to solve for the velocity, since that adds an extra step and you don't actually care about the time of flight.
Again, the reason you have to solve the ODEs is because the entire dynamics of the system are important. 
When I did this, I got $v_f=-v_t \tanh(\cosh^{-1}(\sec(\tan^{-1}(\frac{v_0}{v_t}))))$ and simplified using trig identities and their hyperbolic analogues.
If this is homework, you can shorten the problem considerably by simply saying, "I got this formula from the book" in referring to max height/the ODE solutions.
