The Law of Conservation of Energy and Time

Question

The Law of Conservation of Energy essentially states that in a given closed system, the total energy at any instance of time should be equal. This includes the kinetic energy, gravitational potential energy, elastic potential energy, etc. I noticed that the element of time does not exist in the formula of any of these energies, except maybe the kinetic energy, which contains velocity, which is displacement over time. I have been wondering how one could incorporate time into the Law of Conservation of Energy.

For the sake of argument, say that a rocket with mass, m, is launched from the surface of earth with an initial velocity, v_i. Given gravitational constant, G, radius of earth, R_E, and mass of earth, M_E. The final speed, v_f, of the rocket at any height, h, as well as the maximum height reached by the rocket, h_max can be calculated using the following formula:

$K_i + U_{gi} = K_f + U_{gf} = U_{gmax}$

$\frac{1}{2}mv_i^2 - \frac{GM_Em}{R_E} = \frac{1}{2}mv_f^2 - \frac{GM_Em}{R_E+h} = - \frac{GM_Em}{R_E+h_{max}}$
(edit: thanks for pointing my mistake out, Mateus. It is supposed to be a negative.)

This formula, however, does not tell the time at a specific height. So, how would one go about finding the time elapsed after, for example, the rocket has reached its maximum height?

Answer 1

There's a little mistake in the signs, since the potential energy is given by $U=-\dfrac{GMm}{r}$. The problem of finding the time elapsed between two points of the trajectory may be done by realizing that at any point $$\frac{1}{2}m\dot{x}^2-\dfrac{GMm}{x}=E=-\dfrac{GMm}{H},$$ where $H$ is the maximum height of the particle. Hence $$\frac{dx}{dt}=\sqrt{2GM\left(\dfrac{1}{x}-\dfrac{1}{H}\right)} \implies \\ t=\int_{x_0}^{x_f}\frac{dx}{\sqrt{2GM\left(\dfrac{1}{x}-\dfrac{1}{H}\right)}}=\frac{1}{\sqrt{2GM}}\int_{x_0}^{x_f}\sqrt{\frac{Hx}{H-x}}dx$$ the integral can be performed to yield $$t=\sqrt{\frac{H}{2GM}}\left.\left[\sqrt{x(H-x)}-H\arcsin\sqrt{\frac{x}{H}}\right]\right|_{x=x_0}^{x_f}$$ Substituting the values for $x_0$ and $x_f$, gives the time elapsed between these two points.