Question

1. use mathematical formula and/or diagrams to explain the law of conservation of momentum.

Explanation:

Relate Newton's Third Law to interacting forces

$$ \vec{F}_{12} = -\vec{F}_{21} $$

Apply Newton's Second Law in terms of momentum

$$ \vec{F}_{12} = \frac{d\vec{p}_1}{dt}, \quad \vec{F}_{21} = \frac{d\vec{p}_2}{dt} $$

Integrate to show total momentum conservation

$$ \frac{d\vec{p}_1}{dt} + \frac{d\vec{p}_2}{dt} = 0 \implies \frac{d}{dt}(\vec{p}_1 + \vec{p}_2) = 0 \implies \vec{p}_{\text{total}} = \text{constant} $$

Answer:

According to Newton's Third Law, when two bodies interact in an isolated system, the force exerted by body 1 on body 2 (\(\vec{F}_{21}\)) is equal in magnitude and opposite in direction to the force exerted by body 2 on body 1 (\(\vec{F}_{12}\)):

$$ \vec{F}_{12} = -\vec{F}_{21} $$

Using Newton's Second Law, force is defined as the rate of change of momentum (\(\vec{F} = \frac{d\vec{p}}{dt}\)):

$$ \frac{d\vec{p}_1}{dt} = -\frac{d\vec{p}_2}{dt} $$

Rearranging this equation gives:

$$ \frac{d\vec{p}_1}{dt} + \frac{d\vec{p}_2}{dt} = 0 \implies \frac{d}{dt}(\vec{p}_1 + \vec{p}_2) = 0 $$

Since the time derivative of the total momentum is zero, the total momentum of the system remains constant:

$$ \vec{p}_1 + \vec{p}_2 = \text{constant} $$