This problem involves an inelastic collision, where two objects stick together after colliding. To solve for the final speed, we can use the principle of conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.

Step-by-step solution:

1. Momentum before the collision:

   • The momentum of the moving clay model is given by: $p_{\text{initial}} = m_1 \cdot v_1$ where: $m_1 = 251 \, \text{g} = 0.251 \, \text{kg}$ $v_1 = 67 \, \text{cm/s} = 0.67 \, \text{m/s}$ The second clay model is at rest, so its initial velocity $v_2$ is 0, and hence its initial momentum is zero.

2. Total momentum before the collision: $p_{\text{total\_before}} = (m_1 \cdot v_1) + (m_2 \cdot v_2) = 0.251 \cdot 0.67 + 0 = 0.16817 \, \text{kg} \cdot \text{m/s}$

3. Momentum after the collision: After the collision, the two clay models stick together, so their combined mass is: $m_{\text{combined}} = m_1 + m_2 = 251 \, \text{g} + 313 \, \text{g} = 564 \, \text{g} = 0.564 \, \text{kg}$ Let the final speed of the combined system be $v_f$. Using the conservation of momentum: $p_{\text{total\_before}} = p_{\text{total\_after}} = m_{\text{combined}} \cdot v_f$ Substituting the known values: $0.16817 = 0.564 \cdot v_f$

4. Solving for $v_f$: $v_f = \frac{0.16817}{0.564} = 0.298 \, \text{m/s}$ Converting back to cm/s: $v_f = 0.298 \times 100 = 29.8 \, \text{cm/s}$

Final Answer:

The final speed of the combined clay models is 29.8 cm/s.

Would you like more details or have any questions?

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Here are 5 related questions for further exploration:

1. What would happen if the two clay models did not stick together?
2. How would the final speed change if the second clay model had a larger mass?
3. What is the kinetic energy lost during the collision?
4. Can you derive the formula for the final velocity in an inelastic collision?
5. How does the momentum principle apply in other physical systems?

Tip: In inelastic collisions, kinetic energy is not conserved, but momentum is always conserved.