In a spring gun having spring constant 100 N/m 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/s²).

A ball of mass 4 kg, moving with a velocity of 10 ms^$$-$$1, collides with a spring of length 8 m and force constant 100 Nm^$$-$$1. The length of the compressed spring is x m. The value of x, to the nearest integer, is ____________.

As shown in the figure, a particle of mass 10 kg is placed at a point A. When the particle is slightly displaced to its right, it starts moving and reaches the point B. The speed of the particle at B is x m/s. (Take g = 10 m/s²)

The value of 'x' to the nearest integer is __________.

The potential energy (U) of a diatomic molecule is a function dependent on r (interatomic distance) as

$$U = {\alpha \over {{r^{10}}}} - {\beta \over {{r^5}}} - 3$$

where, $$\alpha$$ and $$\beta$$ are positive constants. The equilibrium distance between two atoms will be $${\left( {{{2\alpha } \over \beta }} \right)^{{a \over b}}}$$, where a = ___________.