1. Given data

• Mass of ball: $m = 4\,\text{kg}$
• Initial speed of ball: $v = 10\,\text{m s}^{-1}$
• Spring constant: $k = 100\,\text{N m}^{-1}$
• Natural length of spring: $L = 8\,\text{m}$

We are asked to find the compressed length of the spring, say $x$ m.

2. Use conservation of mechanical energy

When the ball compresses the spring to the maximum extent, its velocity becomes zero. So, the initial kinetic energy of the ball is completely converted into elastic potential energy of the spring.

$\frac{1}{2}mv^2 = \frac{1}{2}k(\Delta l)^2$

where $\Delta l$ is the compression of the spring.

3. Calculate the initial kinetic energy

$\frac{1}{2}mv^2 = \frac{1}{2}(4)(10)^2 = 2 \cdot 100 = 200\,\text{J}$

4. Equate with spring potential energy

$\frac{1}{2}(100)(\Delta l)^2 = 200$

$50(\Delta l)^2 = 200$

$(\Delta l)^2 = 4$

$\Delta l = 2\,\text{m}$

5. Find the compressed length

Original length of spring $= 8\,\text{m}$. Compression $= 2\,\text{m}$.

So compressed length,

$x = 8 - 2 = 6\,\text{m}$

6. Final answer

To the nearest integer,

$\boxed{6}$

7. Comparison with stored correct answer

Stored correct answer = $6$

Our derived answer also = $6$. So they agree.