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This is a problem involving an **inelastic collision**. When two objects stick together after colliding, the principle of **conservation of momentum** applies. Here's how to solve it step by step:

### Given:
- Mass of the first cart, \( m_1 = 3.00 \, \text{kg} \)
- Initial velocity of the first cart, \( v_1 = +4.00 \, \frac{\text{m}}{\text{s}} \)
- Mass of the second cart, \( m_2 = 5.00 \, \text{kg} \)
- Initial velocity of the second cart, \( v_2 = +2.00 \, \frac{\text{m}}{\text{s}} \)
- The carts stick together after the collision, so their final velocities are the same.

### To Find:
- The final velocity, \( v_f \), of the carts after colliding.

### Step 1: Apply Conservation of Momentum
The total momentum before the collision equals the total momentum after the collision.

\[
m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f
\]

### Step 2: Solve for \( v_f \)
\[
v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}
\]

### Plug in the given values:
\[
v_f = \frac{(3.00 \, \text{kg})(4.00 \, \frac{\text{m}}{\text{s}}) + (5.00 \, \text{kg})(2.00 \, \frac{\text{m}}{\text{s}})}{3.00 \, \text{kg} + 5.00 \, \text{kg}}
\]
\[
v_f = \frac{12.00 \, \text{kg} \cdot \frac{\text{m}}{\text{s}} + 10.00 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}}{8.00 \, \text{kg}}
\]
\[
v_f = \frac{22.00 \, \text{kg} \cdot \frac{\text{m}}{\text{s}}}{8.00 \, \text{kg}}
\]
\[
v_f = 2.75 \, \frac{\text{m}}{\text{s}}
\]

### Final Answer:
The speed of the carts after colliding is \( 2.75 \, \frac{\text{m}}{\text{s}} \).

Do you need further clarification on this solution?

Here are some related questions to deepen your understanding:

1. What would happen if the second cart had a negative velocity?
2. How does the concept of kinetic energy apply to this inelastic collision?
3. How do we differentiate between elastic and inelastic collisions?
4. What role does momentum play in systems with more than two objects?
5. How would the final velocity change if the masses of the carts were different?

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

A 3.00-kg cart with an initial velocity of +4.00 m/s collides inelastically with a 5.00-kg cart with an initial velocity of +2.00 m/s. The carts stick together when they collide. What is their speed after colliding?

Solve an inelastic collision problem involving a 3.00-kg cart and a 5.00-kg cart. Apply the principle of conservation of momentum to find their final speed after colliding. This solution is ideal for high school students studying physics and momentum conservation.

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