In a spring gun having spring constant 100 N m$$^{-1}$$ a small ball $$B$$ of mass 100 g is put in its barrel (as shown in figure) by compressing the spring through 0.05 m. There should be a box placed at a distance $$d$$ on the ground so that the ball falls in it. If the ball leaves the gun horizontally at a height of 2 m above the ground. The value of $$d$$ is ___ m.
$$(g = 10$$ m s$$^{-2})$$

By conservation of mechanical energy: $$\frac{1}{2}kx^2 = \frac{1}{2}mv^2$$

$$\implies v = x\sqrt{\frac{k}{m}} = 0.05 \times \sqrt{\frac{100}{0.1}} = 0.05 \times \sqrt{1000} = 0.05 \times 10\sqrt{10} = 0.5\sqrt{10}\ \text{m/s}$$

$$d = v \times t = v\sqrt{\frac{2h}{g}}$$

$$\implies d = 0.5\sqrt{10} \times \sqrt{\frac{2 \times 2}{10}} = 0.5\sqrt{10} \times \frac{2}{\sqrt{10}} = 0.5 \times 2 = 1\ \text{m}$$