A body is projected vertically upwards from the ground in a medium that produces a resistance force per unit mass of $\displaystyle kv^2 $, where v is the velocity and k is a positive constant. Given that the maximum height is $\displaystyle H = \frac{1}{2k}ln (1+\frac{k(v_0)^2}{g}) $ Show that to double the maximum height, $\displaystyle v_0 $ must be increased by a factor of $\displaystyle (e^{2kH} + 1)^\frac{1}{2} $

where $\displaystyle v_0 $ is the initial velocity

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