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

Q: 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

(a) 6 m/s

(b) 12 m/s

(c) 18 m/s

(d) 14 m/s

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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