# Kepler and Gravitation

### Problems for Kepler's Second Law

Problem : What is the angular momentum of Mercury when it is located at $\vec{r} = (45 \times 10^6 \rm{km}, 57 \times 10^6 \rm{km}, 0)$ relative to the sun and has velocity $\vec{v} = (140 \rm{m/s}, 125 \rm{m/s}, 0)$, and a mass $m = 3.30 \times 10^{23}$ kg?

Problem : Use Kepler's Second Law to explain why planets travel faster near the sun.

Problem : If an Inter-Continental Ballistic Missile (ICBM) is launched into an elliptical path, where in its trajectory will it be traveling the slowest?

Problem : Mercury has an aphelion distance of $69.8 \times 10^6$ kilometers and perihelion distance of $45.9 \times 10^6$ kilometers. What is the ratio $\frac{v_{a}}{v_p}$ where $v_a$ and $v_p$ are the speeds at the apogee and perigee respectively?

Problem : Beginning with $\frac{dA}{dt} = \frac{L}{2m}$, which is just an expression of Kepler's Second Law, prove Kepler's Third Law. Use the facts that $A$, the area of an ellipse, is equal to $\pi ab$ and that the semimmajor axis length is given by $a = \frac{L^2}{GMm^2(1-\epsilon^2)}$.

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