Physics - Gravitation Question with Solution | TestHub

$\frac{-G{mM}_s}{r_1}+\frac{1}{2}{\mathrm{mv}}_1^2=\frac{-Gm{M}_s}{r_2}+\frac{1}{2}{\mathrm{mv}}_2^2$



At point A,

Centripetal force$\frac{{\mathrm{mv}}_1^2}{r_1}=\frac{Gm{M}_s}{r_1^2}$

or$\frac{1}{2}{\text{mv}}_1^2=\frac{GmM{}_s}{r{}_1}$.... (i)

∴  Total energy, E at$A=\frac{1}{2}{\text{mv}}_1^2-\frac{Gm{M}_{\mathrm{s}}}{r_1}$

∴$E=-\frac{Gm{M}_s}{{2r}_2}$....(ii)

From the property of ellipse,

$r_1 + r_2 = 2a$

From eqn. (i) and (ii), we get

$E(r_1 + r_2) = – G{M}_{s}m$

or$E\times 2a=-G{M}_{s}m$

∴$E=-\frac{G{M}_{s}m}{2a}$