## Reduction to One Body Problem

Let $m_1$ be the mass of the satellite, $m_2$ be the mass of Jupiter, $r$ be the vector from the satellite to the planet, and $\theta$ be the angle between the satellite and Jupiter where $\theta_0=\pi/2.$ We will proceed to use polar coordinates in our calculations. The kinetic energy and potential energy of the system are $T=\frac{1}{2}m_1(\dot{r}^2+r^2\dot{\theta}^2)$ and $V=-\frac{Gm_1m_2}{r}$ respectively. Therefore, the Lagrangian is $\mathcal{L}=\frac{1}{2}m_1(\dot{r}^2+r^2\dot{\theta}^2)+\frac{Gm_1m_2}{r}$. Applying the Euler-Lagrange equation to the $r$ coordinate, we have

Applying the Euler-Lagrange equation on the $\theta$ coordinate, we have

From equation $1$, we have that $m_1\ddot{r}-m_1r\dot{\theta}^2+\frac{Gm_1m_2}{r^2}=0$. Note that since angular momentum is conserved in this system, $m_1 r^2\dot{\theta}= l$. Substituting this into equation $1$, we get $m_1\ddot{r}-\frac{l^2}{m_1r^3}+\frac{Gm_1m_2}{r^2}=0$. Integrating with respect to time, we have

which is a statement of conservation of energy. Solving in terms of $\dot{r}=\frac{dr}{dt}$, we have

Note that this is equivalent to

Rewriting the conservation of angular momentum, we have $dt=\frac{m_1r^2 d\theta}{l}.$ Substituting this into equation $4$, we have

Integrating both sides, we have

where $\theta_0$ is the initial angle. Evaluating this integral, we have

We can rewrite this as

Recognizing this as the general equation of a conic, we know that the eccentricity of the orbit is determined by

## Solution of 2001 AP ® PHYSICS C: MECHANICS FREE-RESPONSE QUESTION Mech 2. Part C

(i) If the speed of the satellite is slightly faster, note that the eccentricity will be $% $, implying an elliptical orbit. Additionally, note that $% $. If the satellite is slightly faster, $E$ will increase and thus $r$ will increase except for when $\theta=\theta_0+2\pi n$, where $n\in \mathbb{Z}$, that is, when the satellite passes the injection point.

(ii) If the speed of the satellite is slightly slower, note that the eccentricity will be $% $, implying an elliptical orbit. Additionally, note that $% $. If the satellite is slightly slower, $E$ will decrease and thus $r$ will decrease except for when $\theta=\theta_0+2\pi n$, where $n\in \mathbb{Z}$, that is, when the satellite passes the injection point.