A space vehicle moving in a circular orbit of radius $r_1$transfer to a larger circular orbit of radius $r_2$ by means of an elliptical path between A and B. The transfer is accomplished by a burst of speed $∆V_A$ at A and a second burst of speed  ${\mathrm{\Delta V}}_{\mathrm{B}}$ at B. Then match the following : (R is the radius of the earth, g is the acceleration due to gravity on the surface of the earth, neglect the spin rotation of the earth)

 $\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}$  (A) For circular orbit of radius $r_1$

${\mathrm{V}}_1=\sqrt{\frac{\mathrm{GM}}{{\mathrm{r}}_1}}\Rightarrow {\mathrm{V}}_1=\mathrm{R}\sqrt{\frac{\mathrm{g}}{{\mathrm{r}}_1}}$

From earth, for elliptical orbit A is perihelion
and B is aphelion

From LCAM ,${\mathrm{mV}}_{\mathrm{A}}{\mathrm{r}}_1={\mathrm{mV}}_{\mathrm{B}}{\mathrm{r}}_2$

$\Rightarrow {\mathrm{V}}_{\mathrm{A}}{\mathrm{r}}_1={\mathrm{V}}_{\mathrm{B}}{\mathrm{r}}_2\cdots \cdots$(1)

Apply LCE, ${(T.E)}_A={(T.E)}_B$

$\frac{-\mathrm{GMm}}{{\mathrm{r}}_1}+\frac{1}{2}{\mathrm{mV}}_{\mathrm{A}}^2=\frac{-\mathrm{GMm}}{{\mathrm{r}}_2}+\frac{1}{2}{\mathrm{mV}}_{\mathrm{B}}^2$....... (2)

Solving (1) and (2)

${\mathrm{V}}_{\mathrm{A}}=\sqrt{\frac{2{\mathrm{GMr}}_2}{{\mathrm{r}}_1\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}\mathrm{\&}{\mathrm{V}}_{\mathrm{B}}=\sqrt{\frac{2{\mathrm{GMr}}_1}{{\mathrm{r}}_2\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}$

${\mathrm{V}}_{\mathrm{A}}=\mathrm{R}\sqrt{\frac{2{\mathrm{gr}}_2}{{\mathrm{r}}_1\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}\mathrm{\&}{\mathrm{V}}_{\mathrm{B}}=\mathrm{R}\sqrt{\frac{2{\mathrm{gr}}_1}{{\mathrm{r}}_2\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}$

for circular orbit of radius

${\mathrm{r}}_2{\mathrm{V}}_2=\sqrt{\frac{\mathrm{GM}}{{\mathrm{r}}_2}}=\mathrm{R}\sqrt{\frac{\mathrm{g}}{{\mathrm{r}}_2}}$

$\therefore$The increment in the velocity is

$\mathrm{\Delta V}={\mathrm{V}}_2-{\mathrm{V}}_{\mathrm{B}}=\mathrm{R}\sqrt{\frac{\mathrm{g}}{{\mathrm{r}}_2}}-\mathrm{R}\sqrt{\frac{2{\mathrm{gr}}_1}{{\mathrm{r}}_2\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}$

$\mathrm{\Delta V}=\mathrm{R}\sqrt{\frac{\mathrm{g}}{{\mathrm{r}}_2}}\left[1-\sqrt{\frac{2{\mathrm{r}}_1}{\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}}\right]$