Question

1. how do the directions of these two post-explosion momentum vectors compare?

the two posts explosion momentum vectors have exactly opposite directions one points in the negative direction and the other points in the positive direction

1. use your answer to question #4, your answer to question #5, and the fact that momentum is a vector quantity to determine the post-explosion momentum of the system of two carts (i.e., the vector sum of the momentum of the two carts).

1. use your answers to question #3 and question #6 to explain how one can say that system momentum is conserved in an explosion?

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"direct_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Conservation of Momentum",
"Explosion System",
"Explosion Momentum"
]
}
</pre_analysis>

<reasoning>

Determine total post-explosion system momentum

\[

$$\begin{aligned} &\vec{p}_{\text{total, post}} = \vec{p}_1 + \vec{p}_2\\ &p_1 = -p_2 \implies \vec{p}_{\text{total, post}} = 0\text{ kg}\cdot\text{m/s} \end{aligned}$$

\]

Compare initial and final system momentum

\[

$$\begin{aligned} &\vec{p}_{\text{total, pre}} = 0\text{ kg}\cdot\text{m/s}\\ &\vec{p}_{\text{total, post}} = 0\text{ kg}\cdot\text{m/s}\\ &\vec{p}_{\text{total, pre}} = \vec{p}_{\text{total, post}} \end{aligned}$$

\]
</reasoning>

<answer>

Question 6

The post-explosion momentum of the system of two carts is \(0\text{ kg}\cdot\text{m/s}\). Since the two carts have equal momentum magnitudes but point in exactly opposite directions, their vector sum is zero.

Question 7

System momentum is conserved because the total momentum of the system before the explosion is equal to the total momentum of the system after the explosion (both are equal to \(0\text{ kg}\cdot\text{m/s}\)).
</answer>

<post_analysis>
{
"subject": "Natural Science",
"question_type": "Free Response",
"knowledge_point": [
"Natural Science",
"Physics",
"Conservation of Momentum"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"direct_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Conservation of Momentum",
"Explosion System",
"Explosion Momentum"
]
}
</pre_analysis>

<reasoning>

Determine total post-explosion system momentum

\[

$$\begin{aligned} &\vec{p}_{\text{total, post}} = \vec{p}_1 + \vec{p}_2\\ &p_1 = -p_2 \implies \vec{p}_{\text{total, post}} = 0\text{ kg}\cdot\text{m/s} \end{aligned}$$

\]

Compare initial and final system momentum

\[

$$\begin{aligned} &\vec{p}_{\text{total, pre}} = 0\text{ kg}\cdot\text{m/s}\\ &\vec{p}_{\text{total, post}} = 0\text{ kg}\cdot\text{m/s}\\ &\vec{p}_{\text{total, pre}} = \vec{p}_{\text{total, post}} \end{aligned}$$

\]
</reasoning>

<answer>

Question 6

The post-explosion momentum of the system of two carts is \(0\text{ kg}\cdot\text{m/s}\). Since the two carts have equal momentum magnitudes but point in exactly opposite directions, their vector sum is zero.

Question 7

System momentum is conserved because the total momentum of the system before the explosion is equal to the total momentum of the system after the explosion (both are equal to \(0\text{ kg}\cdot\text{m/s}\)).
</answer>

<post_analysis>
{
"subject": "Natural Science",
"question_type": "Free Response",
"knowledge_point": [
"Natural Science",
"Physics",
"Conservation of Momentum"
]
}
</post_analysis>