A planet orbits the sun in an elliptical path | vedclass

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(A) According to Kepler's second law,the angular momentum of the planet about the sun remains constant throughout its orbit.The angular momentum $L$ i

Vedclass · Accepted Answer

$\frac{r_P}{r_A} = \frac{v_A}{v_P}$

Vedclass · Answer

(A) According to Kepler's second law,the angular momentum of the planet about the sun remains constant throughout its orbit.The angular momentum $L$ is given by $L = mvr \sin(	heta)$. At perihelion $(P)$ and aphelion $(A)$,the velocity vector is perpendicular to the position vector,so $	heta = 90^\circ$ and $\sin(90^\circ) = 1$.Thus,the angular momentum at perihelion is $L_P = m v_P r_P$ and at aphelion is $L_A = m v_A r_A$.Since $L_P = L_A$,we have:$m v_P r_P = m v_A r_A$Dividing both sides by $m$ and rearranging the terms,we get:$v_P r_P = v_A r_A$$\frac{r_P}{r_A} = \frac{v_A}{v_P}$

Column-$I$	Column-$II$
$(1)$ Kepler's first law	$(a)$ Law of periods
$(2)$ Kepler's second law	$(b)$ Law of orbits
$(3)$ Kepler's third law	$(c)$ Law of areas

$A$ planet orbits the sun in an elliptical path as shown in the figure. Let $v_P$ and $v_A$ be the speed of the planet when at perihelion and aphelion respectively. Which of the following relations is correct?

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Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Kepler's first law $(a)$ Law of periods

$(2)$ Kepler's second law $(b)$ Law of orbits

$(3)$ Kepler's third law $(c)$ Law of areas

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