Two balls A and B are thrown vertically upwards from the same location on the surface of the earth with velocities `2 sqrt((gR)/(3)) ` and `sqrt((2gR)/(3))` respectively, where R is the radius of the earth and g is the acceleration due to gravity on the surface of the earth. The ratio of the maximum height attained by A to that attained by B is

Options

A. 2
B. 4
C. 8
D. `4 sqrt2`

Answer

B

Explanation

If h is the maximum height attained , then by the law of conservation of energy
`1/2 mv^2 - (GMm)/( R) = - (GMm)/((R + h))`
or ` v^2 = (GMh)/(R(R + h)) = (2ghR)/((R+h)) " " ( because g = (GM)/(R^2))`
For ball A `, v = 2 sqrt((gR)/(3)) , h = h_A`
` therefore (4gR)/(3) = (2gh_A R)/((R + h_A)) " or " h_A = 2R`
For ball B, v ` = sqrt((2gR)/(3) ) , h = h_B`
`therefore (2gR)/(3) = (2gh_B R)/((R + h_B)) " or " h_B = R/2 therefore h_A/h_B = 4`