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Let the speed of the planet at the perih...

Let the speed of the planet at the perihelion P in figure .(a) be `v_p` and the Sun - planet distance SP be `r_p` . Relate `{r_p, v_p}` to the corresponding quantities at the apphelion `{r_(A) , v_A}` . Will the planet take equal take equal times to traverse BAC and CPB ?

Text Solution

Verified by Experts

`implies` Suppose angular momentum of a planet at P is `L_p` .
`:. L_p = m_(p) r_(p) v_(p)` , where mass of planet is `m_p`
Because angle between `vecr_p and vecv_p` is `90^@` and similarly angular momentum at point A,
`L_(A) = m_(p) r_(A) v_(A)`
`:. m_(p) r_(p) v_(p) = m_(p) r_(A) v_(A)`
`:. (v_p)/(v_(A) )= r_A/r_p`
but `r_A gt r_p :. v_p gt v_A`
The area SBAC bounded by the ellipse and the radius vectors `vec(SB) and vec(SC)` is larger then SBPC. Form Kepler.s second law, equal areas are swept in equal times. Hence , the planet will take a longer time to traverse BAC and CPB.
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Let the speed of the planet at the perihelion P in figure be v_(P) and the Sun planet distance SP be r_(P) . Relater r_(P), v_(P) to the corresponding quantities at the aphelion (r_(A),v_(A)) . Will the planet take equal times to transverse BAC and CPB ?

Let the speed of the planet at the perihelion pin be vP and the Sun-planet distance SP be r_(P) . Relate {r_(P), v_(P)} to the corresponding quantities at the aphelion {r_(A), v_(A)} . Will the planet take equal times to traverse BAC and CPB ?

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