Question

(b) Let r(t) = X(t)i+Y(t)j be the position of the planet at the instant t and we suppose that the sun is located at the origin (0,0). Between the times t; and t2, the line joining the sun and the planet sweeps out an area Altı, t2) (see the blue region). Express A(t1, t2) in terms of X(t), Y(t), X(t)' and Y (t)'. (c) We denote by F(t) the force exerted on the planet by the sun at the instant t. According to Newton's first law of motion, we know that F(t) = m r(t)", (1) where m is the mass of the planet. Moreover, Newton's law of universal gravitation states that GMm F(t) u, (2) || r(t) ||2º where M is the mass of the sun, G is the gravitational constant and u r(t) Deduce |ro| from equations (1) and (2) that the integrand in (6) is a constant function. (d) Conclude that A(t1, t2) depends only on At = t2-t1, hence Kepler's second law holds. =

Use Greens theorem

Answer 1

Page No. (b) FCH) X(t) i + Y(u); Altigt2) dx dy By Green's theorem & Ydx - xdy sfer -34* ) aray 2x 2 If andre :: Altitz) - H 0 YdX - X dy I $ ${re) (t) x(!) – X (UYUL DE Altitz) = 2 C C where curve joining sun to r(t) & then ř (t2) -

Date: Page No. (c) É (+) mr ř" (+) © E(+2 GMm IT (412 IF(t) r () 2 7 i XF GMm 1 F (+)]]> Nxn = ITO : rx Š = 0 from @ om 0 F XF Fxrl = U pxń" || so - (xi) r x r t řxř" dt d (1x) = dt & r(t) x r(t) r (t) x r'ct) = const + (x(+) { + y (4) j) x ( x'ch î + Y Cb9) с xC) C) Y(H) x (4) i = X (H) y (H) Y(H) x[+] =C

X (H) y'e Ylų x'(4) = const * { integrand X(+) y'll - YH X'l is consta . A (t1+2) = $ (3 (69y Ce (t) Y ) Y HY x'/4) ot const function of at = tziti Kepler's law holds . .