The mass of planet Jupiter $1.9\times {10}^{27}$ kg and that of the sun is$1.99\times {10}^{30} kg$. The mean distance of jupiter from the sun is $7.8\times {10}^{11}m$. Assuming that Jupiter moves in circular orbit around the sun, than the speed of Jupiter is ________ $\times {10}^4 m/sec.$

Here  ${\mathrm{M}}_{\mathrm{J}}=1.9\times {10}^{27}\mathrm{kg}$

${\mathrm{M}}_{\mathrm{S}}=1.99\times {10}^{30}\mathrm{kg}$

$\mathrm{r}=7.8\times {10}^{11}\mathrm{m}$

$\mathrm{G}=6.67\times {10}^{-11}{\mathrm{Nm}}^2{\mathrm{kg}}^{-2}$

F = ?

Now $\mathrm{F}=\frac{{\mathrm{GM}}_{\mathrm{J}}{\mathrm{M}}_{\mathrm{S}}}{{\mathrm{r}}^2}$

$=\frac{6.67\times {10}^{-11}\times 1.9\times {10}^{27}\times 1.99\times {10}^{30}}{{(7.8\times {10}^{11})}^2}$

$=4.15\times {10}^{23}\mathrm{N}$

Since the gravitational pull of the sun on Jupiter provides the required centripetal force to Jupiter, therefore the velocity of Jupiter V can be given by the relation,

$\mathrm{F}=\frac{{\mathrm{M}}_{\mathrm{J}}{\mathrm{V}}^2}{\mathrm{r}}$

$\mathrm{V}=\sqrt{\frac{\mathrm{Fr}}{{\mathrm{M}}_{\mathrm{J}}}}=\sqrt{\frac{4.15\times {10}^{23}\times 7.8\times {10}^{11}}{1.9\times {10}^{27}}}$

$\mathrm{V}=1.3\times {10}^4{\mathrm{ms}}^{-1}$