Question

1. consider three trials above. pick whatever three you wish as long as the ratio of \\(m_{\text{red}}/m_{\text{blue}}\\) is different for the three that you pick. for each trial, calculate the post-explosion momentum of each individual cart.

show your work in the table cells.

\

$$\begin{tabular}{|c|c|c|} \\hline trial & post-explosion momentum of red cart (\\text{kg} * \\text{m/s}) & post-explosion momentum of blue cart (\\text{kg} * \\text{m/s}) \\\\ \\hline 1 & & \\\\ \\hline 2 & & \\\\ \\hline 3 & & \\\\ \\hline \\end{tabular}$$

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

Explanation:

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"Explosion Momentum",
"Conservation of Momentum"
],
"current_concepts": [
"Conservation of Momentum",
"Explosion Momentum",
"Vector Directions"
]
}
</pre_analysis>

<reasoning>

Define the system and conservation of momentum

\[

$$\begin{aligned} &\vec{p}_{\text{initial}} = 0 \\ &\vec{p}_{\text{final}} = \vec{p}_{\text{red}} + \vec{p}_{\text{blue}} = 0 \\ &\vec{p}_{\text{red}} = -\vec{p}_{\text{blue}} \end{aligned}$$

\]

Calculate post-explosion momentum for three different mass ratios

\[

$$\begin{aligned} &\text{Trial 1: } m_{\text{red}} = 1.0\text{ kg},\ m_{\text{blue}} = 1.0\text{ kg},\ v_{\text{red}} = -2.0\text{ m/s},\ v_{\text{blue}} = 2.0\text{ m/s} \\ &p_{\text{red}} = 1.0 \times (-2.0) = -2.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 1.0 \times 2.0 = 2.0\text{ kg}\cdot\text{m/s} \\ \\ &\text{Trial 2: } m_{\text{red}} = 2.0\text{ kg},\ m_{\text{blue}} = 1.0\text{ kg},\ v_{\text{red}} = -1.0\text{ m/s},\ v_{\text{blue}} = 2.0\text{ m/s} \\ &p_{\text{red}} = 2.0 \times (-1.0) = -2.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 1.0 \times 2.0 = 2.0\text{ kg}\cdot\text{m/s} \\ \\ &\text{Trial 3: } m_{\text{red}} = 1.0\text{ kg},\ m_{\text{blue}} = 3.0\text{ kg},\ v_{\text{red}} = -3.0\text{ m/s},\ v_{\text{blue}} = 1.0\text{ m/s} \\ &p_{\text{red}} = 1.0 \times (-3.0) = -3.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 3.0 \times 1.0 = 3.0\text{ kg}\cdot\text{m/s} \end{aligned}$$

\]

Compare the directions of the momentum vectors

\[

$$\begin{aligned} &\vec{p}_{\text{red}} = -\vec{p}_{\text{blue}} \\ &\text{Directions are exactly opposite (one is negative, one is positive).} \end{aligned}$$

\]
</reasoning>

<answer>

Question 4

Trial	Post-Explosion Momentum of Red Cart (\(\text{kg}\cdot\text{m/s}\))	Post-Explosion Momentum of Blue Cart (\(\text{kg}\cdot\text{m/s}\))
2	\(-2.0\)	\(2.0\)
3	\(-3.0\)	\(3.0\)

Question 5

The two post-explosion momentum vectors have exactly opposite directions (one points in the negative direction, and the other points in the positive direction).
</answer>

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"Natural Science",
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Answer:

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"step_cot",
"explain_then_answer",
"quick"
],
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"concepts_used": [],
"new_concepts": [
"Explosion Momentum",
"Conservation of Momentum"
],
"current_concepts": [
"Conservation of Momentum",
"Explosion Momentum",
"Vector Directions"
]
}
</pre_analysis>

<reasoning>

Define the system and conservation of momentum

\[

$$\begin{aligned} &\vec{p}_{\text{initial}} = 0 \\ &\vec{p}_{\text{final}} = \vec{p}_{\text{red}} + \vec{p}_{\text{blue}} = 0 \\ &\vec{p}_{\text{red}} = -\vec{p}_{\text{blue}} \end{aligned}$$

\]

Calculate post-explosion momentum for three different mass ratios

\[

$$\begin{aligned} &\text{Trial 1: } m_{\text{red}} = 1.0\text{ kg},\ m_{\text{blue}} = 1.0\text{ kg},\ v_{\text{red}} = -2.0\text{ m/s},\ v_{\text{blue}} = 2.0\text{ m/s} \\ &p_{\text{red}} = 1.0 \times (-2.0) = -2.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 1.0 \times 2.0 = 2.0\text{ kg}\cdot\text{m/s} \\ \\ &\text{Trial 2: } m_{\text{red}} = 2.0\text{ kg},\ m_{\text{blue}} = 1.0\text{ kg},\ v_{\text{red}} = -1.0\text{ m/s},\ v_{\text{blue}} = 2.0\text{ m/s} \\ &p_{\text{red}} = 2.0 \times (-1.0) = -2.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 1.0 \times 2.0 = 2.0\text{ kg}\cdot\text{m/s} \\ \\ &\text{Trial 3: } m_{\text{red}} = 1.0\text{ kg},\ m_{\text{blue}} = 3.0\text{ kg},\ v_{\text{red}} = -3.0\text{ m/s},\ v_{\text{blue}} = 1.0\text{ m/s} \\ &p_{\text{red}} = 1.0 \times (-3.0) = -3.0\text{ kg}\cdot\text{m/s} \\ &p_{\text{blue}} = 3.0 \times 1.0 = 3.0\text{ kg}\cdot\text{m/s} \end{aligned}$$

\]

Compare the directions of the momentum vectors

\[

$$\begin{aligned} &\vec{p}_{\text{red}} = -\vec{p}_{\text{blue}} \\ &\text{Directions are exactly opposite (one is negative, one is positive).} \end{aligned}$$

\]
</reasoning>

<answer>

Question 4

Trial	Post-Explosion Momentum of Red Cart (\(\text{kg}\cdot\text{m/s}\))	Post-Explosion Momentum of Blue Cart (\(\text{kg}\cdot\text{m/s}\))
2	\(-2.0\)	\(2.0\)
3	\(-3.0\)	\(3.0\)

Question 5

The two post-explosion momentum vectors have exactly opposite directions (one points in the negative direction, and the other points in the positive direction).
</answer>

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"subject": "Natural Science",
"question_type": "Multi-part",
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"Natural Science",
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