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# A body of mass m ascends from the earth's surface with zero initial velocity due to action of two forces as shown. The force $F$ varying with $h$ as $\hat F=-2m \vec {g} (1-ah)$ where a is constant. Find the maximum height atained by the body t

$(a)\;\frac{1}{ga} \quad (b)\;\frac{1}{a} \quad (c)\;\frac{1}{2a} \quad (d)\;\frac{g}{2a}$

$\int \limits_0^H (\hat F+m \vec g) dh=0$ [Where H is maximum height attained]
$\int \limits_0^H [-2mg(1-ah)+mg]dh=0$
$\int \limits_0^H mg(1-2h)dh=0$
$mg H (1-aH)=0$
$H= \large\frac{1}{a}$
Maximum height attained $=\large\frac{1}{a}$