My name is Alexander FufaeV and here I write about:
Conservation of Momentum
Important Formula
$$m_1 \, \class{red}{v_1} ~+~ m_2 \, \class{blue}{v_2} ~=~ m_1 \, \class{red}{v'_1} ~+~ m_2 \, \class{blue}{v'_2}$$
$$\class{red}{v_1} ~=~ \class{red}{v'_1} ~+~ \frac{m_2}{m_1} \, \left( \class{blue}{v'_2} ~-~ \class{blue}{v_2} \right)$$
$$\class{blue}{v_2} ~=~ \class{blue}{v'_2} ~-~ \frac{m_1}{m_2} \, \left( \class{red}{v_1} ~-~ \class{red}{v'_1} \right)$$
$$m_1 ~=~ \frac{ \class{blue}{v'_2} ~-~ \class{blue}{v_2} }{ \class{red}{v_1} ~-~ \class{red}{v'_1} } \, m_2 $$
$$m_2 ~=~ \frac{ \class{red}{v_1} ~-~ \class{red}{v'_1} }{ \class{blue}{v'_2} ~-~ \class{blue}{v_2} } \, m_1$$
$$\class{red}{v'_1} ~=~ \class{red}{v_1} ~-~ \frac{m_2}{m_1} \, \left( \class{blue}{v'_2} ~-~ \class{blue}{v_2} \right)$$
$$\class{blue}{v'_2} ~=~ \class{blue}{v_2} ~+~ \frac{m_1}{m_2} \, \left( \class{red}{v_1} ~-~ \class{red}{v'_1} \right)$$
What do the formula symbols mean?
Velocity of the first body (before)
$$ \class{red}{v_1} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Velocity of the first body before the collision with the second body. The first body has the momentum \( p_1 = m_1 \, v_1 \).
Velocity of the second body (before)
$$ \class{blue}{v_2} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Velocity of the second body before the collision with the first body. The second body has the momentum \( p_2 = m_2 \, v_2 \).
Mass of the first body
$$ m_1 $$ Unit $$ \mathrm{kg} $$
Here it is assumed that the mass of the first body remains the same before and after the collision.
Mass of the second body
$$ m_2 $$ Unit $$ \mathrm{kg} $$
Here it is assumed that the mass of the first body remains the same before and after the collision.
Velocity of the first body (after)
$$ \class{red}{v'_1} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Velocity of the first body after the collision with the second body.
Velocity of the second body (after)
$$ \class{blue}{v'_2} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Velocity of the second body after the collision with the first body.
Conservation of momentum is one of the laws of conservation in physics and applies when no force acts on a particle:
$$ \begin{align} \boldsymbol{F} = 0 \end{align} $$
The Newton's 2nd law states the following:
$$ \begin{align} \boldsymbol{F} = \frac{\text{d}\boldsymbol{p}}{ \text{d}t } \end{align} $$
With Eq. 1
the time derivative of the momentum disappears:
$$ \begin{align} 0 ~=~ \frac{\text{d}\boldsymbol{p}}{ \text{d}t } \end{align} $$
If the pulse does not change over time, it is constant:
$$ \begin{align} \boldsymbol{p} = \boldsymbol{p}_0 = \text{const.} \end{align} $$