Physics - W.E.P. Question with Solution | TestHub

Given values:

Mass of block, $m=2\mathrm{kg}$

Initial speed, $v=4m/s$

Kinetic friction force, $f_k=15\mathrm{N}$

Spring constant, $k=10,000N/m$

Let $x$ be the compression of the spring.

Initial kinetic energy:

$$K{E}_i=\frac{1}{2}m{v}^2$$

Final kinetic energy (when the block is motionless):

$$K{E}_f=0$$

Elastic potential energy stored in the spring:

$$P{E}_s=\frac{1}{2}k{x}^2$$

Work done by friction:

$$W_f=f_{k}x$$

According to the work-energy theorem, the change in kinetic energy is equal to the total work done on the block. Here, the initial kinetic energy is dissipated by the spring and friction.

$$K{E}_i=P{E}_s+W_f$$

$$\frac{1}{2}m{v}^2=\frac{1}{2}k{x}^2+f_{k}x$$

Substitute the given values:

$$\frac{1}{2}(2\mathrm{kg})(4m/s{)}^2=\frac{1}{2}(10,000N/m)x^2+(15\mathrm{N})x$$

$$16=5000x^2+15x$$

Rearrange into a quadratic equation:

$$5000x^2+15x-16=0$$

Using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$:

$$x=\frac{-15\pm \sqrt{15^2-4(5000)(-16)}}{2(5000)}$$

$$x=\frac{-15\pm \sqrt{225+320000}}{10000}$$

$$x=\frac{-15\pm \sqrt{320225}}{10000}$$

$$x=\frac{-15\pm 565.88}{10000}$$

Since compression $x$ must be positive:

$$x=\frac{-15+565.88}{10000}$$

$$x=\frac{550.88}{10000}$$

$$x\approx 0.055088\mathrm{m}$$

Convert to centimeters:

$$x\approx 0.055088\times 100\mathrm{cm}$$

$$x\approx 5.51\mathrm{cm}$$

The spring compresses by approximately $5.5\mathrm{cm}$.

The final answer is $\boxed{5.5\mathrm{cm}}$.