Question $1$: $15$ Points

Using the known properties of an elliptic orbit, determine a relationship between the change in time, $t_2-t_1,$ as

a function of the movement of the spacecraft from an a true $v(t_1)=v_1$ at time $t=t_1$ to a final true anomaly

$v(t_2)=v_2$ and time $t=t_2.$ Express your result in the following form:

$t_2-t_1={\displaystyle \underset{v_1}{\overset{v_2}{}}}f(v,p,e,)dv$

where $p$ is the parameter $($semi$-$latus rectum$),e$ is the eccentricity, and $$ is the gravitational parameter.

Note: the quantities $p$ and $e$ provides key information about the orbit. For this problem it should be possible to

simplify the expression in Eq$.(1)$ such that the function $f$ depends only upon $v,p,e,$ and $.$