Kepler's Laws, impressive as they are, were purely descriptive; Newton's great achieve-ment was

(a) We define the planet's angular momentum as L = r x p, where r is the position vector and p i

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Kepler's Laws, impressive as they are, were purely descriptive; Newton's great achieve-ment was to find an underlying cause for them. In this project, you will derive Kepler's Second Law from Newton's Law of Gravity. Consider a coordinate system centered at the sun.³ Let $vec{r}$ be the position vector of a planet and let $vec{v}$ and $vec{a}$ be the planet's velocity and acceleration, respectively. Define $vec{L} = vec{r} imes vec{v}$. (This is a multiple of the planet's angular momentum.) (a) Show that $frac{dvec{L}}{dt} = vec{r} imes vec{a}$. (b) Consider the planet moving from $vec{r}$ to $vec{r} + Delta vec{r}$. Explain why the area $Delta A$ about the origin swept out by the planet is approximately $frac{1}{2}||Delta vec{r} imes vec{r}||$. (c) Using part (b), explain why $frac{dA}{dt} = frac{1}{2}||vec{L}||$. (d) Newton's Laws imply that the planet's gravitational acceleration, $vec{a}$, is directed toward the sun. Using this fact and part (a), explain why $vec{L}$ is constant. (e) Use parts (c) and (d) to explain Kepler's Second Law. (f) Using Kepler's Second Law, determine whether a planet is moving most quickly when it is closest to, or farthest from, the sun.

          Kepler's Laws, impressive as they are, were purely descriptive; Newton's great achieve-ment was to find an underlying cause for them. In this project, you will derive Kepler's Second Law from Newton's Law of Gravity.
Consider a coordinate system centered at the sun.³ Let $vec{r}$ be the position vector of a planet and let $vec{v}$ and $vec{a}$ be the planet's velocity and acceleration, respectively. Define $vec{L} = vec{r} 	imes vec{v}$. (This is a multiple of the planet's angular momentum.)
(a) Show that $frac{dvec{L}}{dt} = vec{r} 	imes vec{a}$.
(b) Consider the planet moving from $vec{r}$ to $vec{r} + Delta vec{r}$. Explain why the area $Delta A$ about the origin swept out by the planet is approximately $frac{1}{2}||Delta vec{r} 	imes vec{r}||$.
(c) Using part (b), explain why $frac{dA}{dt} = frac{1}{2}||vec{L}||$.
(d) Newton's Laws imply that the planet's gravitational acceleration, $vec{a}$, is directed toward the sun. Using this fact and part (a), explain why $vec{L}$ is constant.
(e) Use parts (c) and (d) to explain Kepler's Second Law.
(f) Using Kepler's Second Law, determine whether a planet is moving most quickly when it is closest to, or farthest from, the sun.

$Kepler's Laws, impressive as they are, were purely descriptive; Newton's great achieve-ment was to find an underlying cause for them. In this project, you will derive Kepler's Second Law from Newton's Law of Gravity. Consider a coordinate system centered at the sun.³ Let vecr be the position vector of a planet and let vecv and veca be the planet's velocity and acceleration, respectively. Define vecL = vecr imes vecv. (This is a multiple of the planet's angular momentum.) (a) Show that fracdvecLdt = vecr imes veca. (b) Consider the planet moving from vecr to vecr + Delta vecr. Explain why the area Delta A about the origin swept out by the planet is approximately frac12||Delta vecr imes vecr||. (c) Using part (b), explain why fracdAdt = frac12||vecL||. (d) Newton's Laws imply that the planet's gravitational acceleration, veca, is directed toward the sun. Using this fact and part (a), explain why vecL is constant. (e) Use parts (c) and (d) to explain Kepler's Second Law. (f) Using Kepler's Second Law, determine whether a planet is moving most quickly when it is closest to, or farthest from, the sun.$

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