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

When the ball collides with the spring and compresses it maximally, all the kinetic energy of the ball is converted into the potential energy stored in the spring. At maximum compression, the ball momentarily comes to rest.

Using energy conservation: $$\frac{1}{2}mv^2 = \frac{1}{2}kx_c^2$$, where $$x_c$$ is the compression of the spring. Substituting the given values: $$\frac{1}{2}(4)(10)^2 = \frac{1}{2}(100)x_c^2$$, which gives $$200 = 50 x_c^2$$, so $$x_c^2 = 4$$ and $$x_c = 2$$ m.

The natural length of the spring is 8 m. After compression by 2 m, the length of the compressed spring is $$x = 8 - 2 = 6$$ m.

The value of $$x$$ is $$6$$.