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, vi. Given gravitational constant, G, radius of earth, RE, and mass of earth, ME. The final speed, vf, of the rocket at any height, h, as well as the maximum height reached by the rocket, hmax 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?
Best Answer
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.