Question

Find the maximum compression in the spring?

Answer 1

$\sqrt{2}$ m (approximately 1.414 m).

Answer 2

Using conservation of energy, the initial kinetic energy of the block is converted into the potential energy of the spring at maximum compression. Thus,

$$\frac{1}{2} M u^2 = \frac{1}{2} k x_m^2.$$

Canceling $\frac{1}{2}$ from both sides and solving for $x_m$:

$$x_m = \sqrt{\frac{M}{k}}\, u.$$

Plugging in the values $M = 2\,\text{kg}$, $k = 100\,\text{N/m}$, and $u = 10\,\text{m/s}$:

$$x_m = \sqrt{\frac{2}{100}} \times 10 = \sqrt{0.02} \times 10 = \sqrt{2} \approx 1.414\,\text{m}.$$

Core Explanation

• Equate kinetic and spring potential energy.
• Solve for $x_m$ to get $x_m = \sqrt{\frac{M}{k}}\, u$.
• Substitute numerical values to obtain $x_m = \sqrt{2} \approx 1.414\,\text{m}$.