**Graphic design is my passion.** A mass m1=m1= 3.0 [kg] with initial velocity v0=v0= 15.0 [m/s] collides with a mass-spring system m2=m2= 1.0 [kg], initially at rest but able to recoil. The spring is massless with spring constant k=k= 100. [N/m]. Assume no friction. What is the maximum compression of the spring when the collision is perfectly inelastic?

1.30 [m]

1.73 [m]

3.81 [m]

5.08 [m]

To find the maximum compression of the spring when the collision is perfectly inelastic, we can follow these steps:

### 1. Conservation of Momentum in the Inelastic Collision
Since the collision is perfectly inelastic, the two masses stick together after the collision. Using the conservation of linear momentum, we have:

\[
m_1 v_0 + m_2 \cdot 0 = (m_1 + m_2) v_f
\]

Where:
- \( m_1 = 3.0 \, \text{kg} \)
- \( v_0 = 15.0 \, \text{m/s} \)
- \( m_2 = 1.0 \, \text{kg} \)

Substitute these values:

\[
3.0 \times 15.0 = (3.0 + 1.0) v_f
\]

\[
45 = 4 v_f
\]

\[
v_f = \frac{45}{4} = 11.25 \, \text{m/s}
\]

### 2. Energy Conservation to Find Maximum Compression
After the collision, the combined mass \( (m_1 + m_2) \) moves with velocity \( v_f \) and compresses the spring until the kinetic energy is fully converted to potential energy in the spring. The potential energy in the spring at maximum compression is given by:

\[
\frac{1}{2} k x^2
\]

And the kinetic energy of the combined mass just after the collision is:

\[
\frac{1}{2} (m_1 + m_2) v_f^2
\]

Setting these equal to each other due to energy conservation:

\[
\frac{1}{2} (m_1 + m_2) v_f^2 = \frac{1}{2} k x^2
\]

Canceling \(\frac{1}{2}\) from both sides and substituting the values:

\[
(3.0 + 1.0) \times (11.25)^2 = 100 \cdot x^2
\]

\[
4 \times 126.5625 = 100 \cdot x^2
\]

\[
506.25 = 100 \cdot x^2
\]

\[
x^2 = \frac{506.25}{100} = 5.0625
\]

\[
x = \sqrt{5.0625} \approx 2.25 \, \text{m}
\]

### Answer
The maximum compression of the spring is approximately \( \mathbf{2.25 \, \text{m}} \), which is not listed in the options provided. The solution should be double-checked in case there is an error in rounding or setup interpretation.

Do you need more details or have further questions?

1. What is a perfectly inelastic collision and how does it differ from elastic collisions?
2. How would the solution change if friction was present?
3. How does spring constant \( k \) affect the maximum compression?
4. How would the solution change if the masses were swapped?
5. Can we solve this problem with an energy approach alone?

**Tip:** Remember that in energy conservation problems, converting units consistently can prevent common mistakes.

A mass m1=3.0 [kg] with initial velocity v0=15.0 [m/s] collides with a mass-spring system m2=1.0 [kg], initially at rest but able to recoil. The spring is massless with spring constant k=100. [N/m]. Assume no friction. What is the maximum compression of the spring when the collision is perfectly inelastic?

Learn to calculate the maximum compression of a spring in a perfectly inelastic collision involving a mass-spring system. This problem applies concepts from conservation of momentum and energy, using key formulas to analyze the interaction between a moving mass and a spring system with a constant of 100 N/m.

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