Three particles are projected vertically upward from a point on the surface of the earth with velocities `sqrt(2gR//3)` surface of the earth. The maximum heights attained are respectively `h_(1),h_(2),h_(3)`.

Options

A. `h_(1):h_(2)=2:3`
B. `h_(2):h_(3)=3:4`
C. `h_(1):h_(3)=1:4`
D. `h_(2)=R`

Answer

C, D

Explanation

Case: I
`U_(i)+K_(f)=U_(f)+K_(g)`
`(-Gm_(e)m)/(R)+(1)/(2)mv^(2)=-(GM_(e)m)/(R+h_(1))+0`
`(-GM_(e)m)/(R)+(1)/(2)m(2GM_(e))/(R^(2))(R)/(3)=-(GM_(e)m)/(R+h_(1))`
`-(1)/(R)+(1)/(3R)=-(1)/(R+h_(1))impliesh_(1)=(R)/(2)`
Case II:
`U_(i)+K_(i)=U_(f)+K_(g)`
`-(GM_(e)m)/(R)+(1)/(2)mv^(2)=-(GM_(e)m)/(R+h_(2))+0`
`(-GM_(e)m)/(R)+(1)/(2)m(2GM_(e))/(R^(2))R=-(GM_(e)m)/(R+h_(2))`
`-(1)/(R)+(1)/(2R)=-(1)/(R+h_(2))impliesh_(2)=R`
Case III:
`U_(i)+K_(i)=U_(f)+K_(g)`
`-(GM_(e)m)/(R)+(1)/(2)m(4GM_(e))/(R^(2))(R)/(3)=-(GM_(e)m)/(R+h_(3))`
`-(1)/(R)+(1)/(3R)=-(1)/(R+h_(3))impliesh_(3)=2R`