A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero ? Mass of the sun = `2xxl0^30 kg`, mass of the earth = `6xxl0^24 kg`. Neglect the effect of other planets etc. (orbital radius = `1.5 xx 10^11 m`).

Explanation

Mass of sun, `M= 2 xx 10^(30)kg`
Mass of earth, `m= 6 xx 10^(24)kg`
Distance between sun and earth, `r= 1.5 xx 10^(11)m`
Let a point P, gravitational force on the rocket due to earth= gravitational force on rocket due to sun i.e., `(Gm)/(x^(2))= (GM)/((r-x)^(2))`
`((r-x)^(2))/((x)^(2))= (M)/(m) rArr ((r-x)/(x))^(2) = (2 xx 10^(30))/(6 xx 10^(24))= (10^(6))/(3)`
`or (r-x)/(x)= (10^(3))/(sqrt3) or (r )/(x)= 1 + (10^(3))/(sqrt3) = (sqrt3+10^(3))/(sqrt3)`
or `x= (r sqrt3)/(sqrt3+ 10^(3)) = (1.5 xx 10^(11) xx sqrt3)/(sqrt3+ 10^(3))`
`therefore x= 2.6 xx 10^(8)m`