A 375 gram cart on an air track approaches another 325 gram cart at v0 = 55.0 cm/s and collides with it, resulting in the two carts sticking together. What is the final speed of the two carts stuck together?

To solve this problem, we can use the principle of conservation of momentum, since there are no external forces acting on the system (assuming an ideal air track with negligible friction).

### Given Data:
- Mass of the first cart, \( m_1 = 375 \, \text{g} = 0.375 \, \text{kg} \)
- Mass of the second cart, \( m_2 = 325 \, \text{g} = 0.325 \, \text{kg} \)
- Initial velocity of the first cart, \( v_1 = 55.0 \, \text{cm/s} = 0.55 \, \text{m/s} \)
- Initial velocity of the second cart, \( v_2 = 0 \, \text{m/s} \) (since it’s not moving)

### Approach:
1. **Conservation of Momentum:**  
   Since the two carts stick together after the collision, this is a perfectly inelastic collision. In an inelastic collision, the total momentum before and after the collision is conserved.

   The total momentum before the collision is:
   \[
   p_{\text{initial}} = m_1 v_1 + m_2 v_2
   \]

   After the collision, the total mass is \( m_1 + m_2 \) and the final velocity is \( v_f \):
   \[
   p_{\text{final}} = (m_1 + m_2) v_f
   \]

   By the conservation of momentum, \( p_{\text{initial}} = p_{\text{final}} \):
   \[
   m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f
   \]

2. **Solve for \( v_f \):**
   Substitute the values:
   \[
   (0.375 \, \text{kg})(0.55 \, \text{m/s}) + (0.325 \, \text{kg})(0 \, \text{m/s}) = (0.375 \, \text{kg} + 0.325 \, \text{kg}) v_f
   \]
   \[
   (0.375 \times 0.55) = (0.7) v_f
   \]
   \[
   0.20625 = 0.7 v_f
   \]
   \[
   v_f = \frac{0.20625}{0.7}
   \]
   \[
   v_f = 0.2946 \, \text{m/s}
   \]

### Final Answer:
The final speed of the two carts stuck together is approximately **0.2946 m/s**.

Would you like further details on the calculation steps or have any other questions?

Here are some related questions you might explore:
1. What would happen to the final velocity if the second cart were moving towards the first cart at 20 cm/s?
2. How much kinetic energy is lost in the collision?
3. What would be the final speed if both carts had the same mass?
4. How would the final speed change if the mass of the first cart were doubled?
5. How can you calculate the change in momentum for each cart individually?

**Tip:** For inelastic collisions where objects stick together, always remember to use the conservation of momentum since kinetic energy is not conserved.

Solve a physics problem involving the collision of two carts on an air track. Using conservation of momentum, calculate the final speed of a 375 gram cart colliding with a 325 gram cart at 55.0 cm/s. This step-by-step solution is ideal for students in Grades 9-12.

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