Question

If orbital velocity of a planet is given by $v = G ^{ a} M ^{ b } R ^{ c }$, then what is the value of $\frac{2 a + b -3 c }{3 b } ?$ [where, $G =$ gravitational constant, $M=$ mass of planet, $R=$ Radius of orbit]

Answer 1

$v =\sqrt{\frac{ GM }{ R }}$
$\Rightarrow a =\frac{1}{2} ; b =\frac{1}{2} ; c =-\frac{1}{2}$
Now, $\frac{2 a + b -3 c }{3 b }=\frac{2\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)-3\left(-\frac{1}{2}\right)}{3\left(\frac{1}{2}\right)}$
$=2$