mrkrk/website
1
0
Fork 0
mirror of https://github.com/grey-cat-1908/website.git synced 2024-11-14 12:07:27 +03:00
Code Issues Projects Releases Packages Wiki Activity Actions

Update index.md

Browse source
This commit is contained in:
Виктор 2024-04-29 15:25:58 +03:00 committed by GitHub
parent bbf2bde26e
commit 48ac7c1e4f
No known key found for this signature in database
GPG key ID: B5690EEEBB952194
1 changed files with 19 additions and 0 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

19
content/blog/1d-collisions/index.md
View file

@ -9,15 +9,20 @@ Consider two particles, designated as particles `1` and `2`, with respective mas
In any collision, momentum is conserved. An elastic collision is an collision, as a result of which the total kinetic energy of the colliding particles is conserved. In any collision, momentum is conserved. An elastic collision is an collision, as a result of which the total kinetic energy of the colliding particles is conserved.
The conservation of momentum before and after the collision is expressed by the following equation:
$$ $$
m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} = m_{1}\vec{v'}_{1} + m_{2}\vec{v'}_{2} m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} = m_{1}\vec{v'}_{1} + m_{2}\vec{v'}_{2}
$$ $$
And the conservation of the total kinetic energy before and after the collision is expressed by the following equation:
$$ $$
\frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2} \frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2}
$$ $$
The aforementioned equations can be transformed into the following system:
$$ $$
\begin{cases} \begin{cases}
m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}}) \\ m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}}) \\
@ -25,22 +30,32 @@ m_{1}({v_{1}}^2 - {v'_{1}}^2) = m_{2}({v'_{2}}^2 - {v_{2}}^2)
\end{cases} \end{cases}
$$ $$
The division of each side of the top equation by each side of the bottom equation yields:
$$ $$
\frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2} \frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2}
$$ $$
The preceding equation may be transformed using the difference of two squares formula to obtain the following:
$$ $$
\frac{v_{1} - v'_{1}}{(v_1 + v'_1)(v_1 - v'_1)} = \frac{v'_{2} - v_{2}}{(v'_2 - v_2)(v'_2 + v_2)} \frac{v_{1} - v'_{1}}{(v_1 + v'_1)(v_1 - v'_1)} = \frac{v'_{2} - v_{2}}{(v'_2 - v_2)(v'_2 + v_2)}
$$ $$
This expression can be transformed as follows:
$$ $$
\frac{1}{v_1 + v'_1} = \frac{1}{v'_2 + v_2} \frac{1}{v_1 + v'_1} = \frac{1}{v'_2 + v_2}
$$ $$
From this expression follows:
$$ $$
v_1 + v'_1 = v'_2 + v_2 v_1 + v'_1 = v'_2 + v_2
$$ $$
From this equation, the following system can be derived:
$$ $$
\begin{cases} \begin{cases}
v'_1 = v_2 + v'_2 - v_1 \\ v'_1 = v_2 + v'_2 - v_1 \\
@ -48,6 +63,8 @@ v'_2 = v_1 + v'_1 - v_2
\end{cases} \end{cases}
$$ $$
The law of conservation of momentum then yields the following:
$$ $$
\begin{cases} \begin{cases}
m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2) \\ m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2) \\
@ -55,6 +72,8 @@ m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2
\end{cases} \end{cases}
$$ $$
From this, we can derive the velocity of the particles following the collision as follows:
$$ $$
\begin{cases} \begin{cases}
v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2} \\ v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2} \\