1. Law of Conservation of Momentum

The law of conservation of momentum states that in a closed system, the total momentum remains constant over time, provided no external forces act on it. This means that the momentum before an interaction, such as a collision, is equal to the momentum after the interaction.

2. Momentum Calculation for the Carts

(a) Total Momentum Before the Collision

To find the total momentum before the collision, we calculate the momentum of each cart and then sum them.

• Momentum of the 2 kg cart: $p_1 = m_1 \times v_1 = 2 \, \text{kg} \times 4 \, \text{m/s} = 8 \, \text{kg m/s} \, (\text{to the right})$

• Momentum of the 3 kg cart: $p_2 = m_2 \times v_2 = 3 \, \text{kg} \times (-2) \, \text{m/s} = -6 \, \text{kg m/s} \, (\text{to the left})$

Now, we add the momenta: $\text{Total momentum} = p_1 + p_2 = 8 \, \text{kg m/s} + (-6 \, \text{kg m/s}) = 2 \, \text{kg m/s}$

(b) Velocity of the Combined Carts After the Collision

After the collision, the two carts stick together, so we can use the conservation of momentum to find their velocity.

Let $V_f$ be the final velocity after the collision. The total momentum after the collision is: $\text{Total momentum} = (m_1 + m_2) \times V_f$

Setting the total momentum before the collision equal to the total momentum after: $2 \, \text{kg m/s} = (2 \, \text{kg} + 3 \, \text{kg}) \times V_f$ $2 \, \text{kg m/s} = 5 \, \text{kg} \times V_f$ $V_f = \frac{2 \, \text{kg m/s}}{5 \, \text{kg}} = 0.4 \, \text{m/s}$

Thus, the combined carts move to the right at 0.4 m/s after the collision.

Summary

• Total momentum before the collision: 2 kg m/s
• Velocity of the combined carts after the collision: 0.4 m/s to the right

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the outcome change if the 3 kg cart were moving at a higher speed?
2. What would happen if the carts did not stick together after the collision?
3. How does the direction of motion affect the calculation of momentum?
4. Can you explain how momentum conservation applies to other types of collisions?
5. What are real-life examples of conservation of momentum in action?

Tip: Always pay attention to the direction of the velocities when calculating momentum; it can greatly affect the results!