A small asteroid is orbiting around the sun in a circular orbit of radius `r_(0)` with speed `V_(0)`. A rocket is launched from the asteroid with speed `V = alpha V_(0)` , where V is the speed relative to the sun. The highest value of `alpha` for which the rocket will remain bound to the solar system is (ignoring gravity due to the asteroid and effects of other planets) –

Options

A. `sqrt2`
B. 2
C. `sqrt3`
D. 1

Answer

A

Explanation

Mechanical energy of asteroid `=1/2mV^2-(GMm)/(r_0)`

Rocket will remain bound to the solar system. If,
`-(GMm)/(r_0)+1/2mV^2=0rArr(GMm)/(r_0)=1/2mV^2rArr(GMm)/(r_0)`
`=1/2malpha^2V_0^2`
`mV_0^2=1/2malpha^2v_0^2rArralpha=sqrt2`