Question
What does the conservation of momentum look like as an equation?
Answers
Answered by
GPT-4o mini
The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. Mathematically, this can be expressed as:
\[
\sum \mathbf{p}_{\text{initial}} = \sum \mathbf{p}_{\text{final}}
\]
Where:
- \(\sum \mathbf{p}_{\text{initial}}\) is the total initial momentum of all objects in the system.
- \(\sum \mathbf{p}_{\text{final}}\) is the total final momentum of all objects in the system.
Momentum \(\mathbf{p}\) for a single object is given by:
\[
\mathbf{p} = m \mathbf{v}
\]
Where:
- \(m\) is the mass of the object.
- \(\mathbf{v}\) is the velocity of the object.
For a system of \(n\) objects, the total momentum can be expressed as:
\[
\sum \mathbf{p} = \sum_{i=1}^{n} m_i \mathbf{v}_i
\]
In a more practical sense, whenever two or more objects interact, the momentum before the interaction equals the momentum after the interaction, which can be written as:
\[
m_1 \mathbf{v}_{1,\text{initial}} + m_2 \mathbf{v}_{2,\text{initial}} = m_1 \mathbf{v}_{1,\text{final}} + m_2 \mathbf{v}_{2,\text{final}}
\]
This illustrates how momentum is conserved in a system over time, provided that no external forces are acting on it.
\[
\sum \mathbf{p}_{\text{initial}} = \sum \mathbf{p}_{\text{final}}
\]
Where:
- \(\sum \mathbf{p}_{\text{initial}}\) is the total initial momentum of all objects in the system.
- \(\sum \mathbf{p}_{\text{final}}\) is the total final momentum of all objects in the system.
Momentum \(\mathbf{p}\) for a single object is given by:
\[
\mathbf{p} = m \mathbf{v}
\]
Where:
- \(m\) is the mass of the object.
- \(\mathbf{v}\) is the velocity of the object.
For a system of \(n\) objects, the total momentum can be expressed as:
\[
\sum \mathbf{p} = \sum_{i=1}^{n} m_i \mathbf{v}_i
\]
In a more practical sense, whenever two or more objects interact, the momentum before the interaction equals the momentum after the interaction, which can be written as:
\[
m_1 \mathbf{v}_{1,\text{initial}} + m_2 \mathbf{v}_{2,\text{initial}} = m_1 \mathbf{v}_{1,\text{final}} + m_2 \mathbf{v}_{2,\text{final}}
\]
This illustrates how momentum is conserved in a system over time, provided that no external forces are acting on it.