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

Two balls A and B are thrown vertically upwards from the same location

1.	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
Answer» 2 4 8 `4 sqrt2` Solution :If h is the MAXIMUM HEIGHT ATTAINED , then by the lawof 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`