A body of mass 8 kg is moved by a force F = 3x N , where x is the distance covered. Initial position is x = 2 m and the final position is x = 10 m. The initial speed is 0.0 m/s. The final speed is

Ans: (a)

Sol: $\displaystyle F = 3 x $ ; (given)

$\displaystyle m v \frac{dv}{dx} = 3 x $

$\displaystyle m v dv = 3 x dx $

$\displaystyle \int_{0}^{v’} m v dv = \int_{2}^{10} 3 x dx $

$\displaystyle m \int_{0}^{v’} v dv = 3 \int_{2}^{10} x dx $

$\displaystyle 8 [\frac{v^2}{2}]_{0}^{v’} = 3 [\frac{x^2}{2}]_{2}^{10} $

$\displaystyle 8 [\frac{v’^2}{2}-0 ] = 3 [\frac{10^2}{2} – \frac{2^2}{2}] $

$\displaystyle v’^2 = 36 $

v’= 6 m/s