diff --git a/content/blog/1d-collisions/index.md b/content/blog/1d-collisions/index.md index 7b3fe76..854eda0 100644 --- a/content/blog/1d-collisions/index.md +++ b/content/blog/1d-collisions/index.md @@ -16,13 +16,13 @@ There are numerous reasons why this project is relevant, as well as a wide range ## 3. The consequences of collisions -Firstly, it is essential to comprehend how the velocity of the particles after collision is calculated. We will solve the problem in the elastic collision model (one-dimensional Newtonian). +Firstly, it is essential to comprehend how the velocity of the bodies after collision is calculated. We will solve the problem in the elastic collision model (one-dimensional Newtonian). -### 3.1 With particle +### 3.1 With body -Consider two particles, designated as particles `1` and `2`, with respective masses $m_1$ and $m_2$. Before the collision, the velocities of particles 1 and 2 are $v_1$ and $v_2$, respectively. After the collision, the velocities of particles 1 and 2 are $v_1'$ and $v_2'$, respectively. +Consider two bodies, designated as bodies `1` and `2`, with respective masses $m_1$ and $m_2$. Before the collision, the velocities of bodies 1 and 2 are $v_1$ and $v_2$, respectively. After the collision, the velocities of bodies 1 and 2 are $v_1'$ and $v_2'$, respectively. -In any collision, momentum is conserved. A collision between to particles is said to be elastic if it involves no change in their internal state. Accordingly, when the law of conservation of energy is applied to such a collision, the internal energy of the particles may be neglected.[^2] +In any collision, momentum is conserved. A collision between to bodies is said to be elastic if it involves no change in their internal state. Accordingly, when the law of conservation of energy is applied to such a collision, the internal energy of the bodies may be neglected.[^2] The conservation of momentum before and after the collision is expressed by the following equation: @@ -87,7 +87,7 @@ m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2 \end{cases} $$ -From this, we can derive the velocity of the particles following the collision as follows: +From this, we can derive the velocity of the bodies following the collision as follows: $$ \begin{cases} @@ -98,11 +98,11 @@ $$ ### 3.2 With wall -Consider particle and a wall with respective masses $m$ and $m_w \rightarrow \infty$. Before the collision, the velocities of the particle and the wall are $v$ and $v_w = 0$, respectively. After the collision, the velocities of the particle and the wall are $v'$ and $v_w'$, respectively. +Consider bodie and a wall with respective masses $m$ and $m_w \rightarrow \infty$. Before the collision, the velocities of the bodie and the wall are $v$ and $v_w = 0$, respectively. After the collision, the velocities of the bodie and the wall are $v'$ and $v_w'$, respectively. In order to proceed, we will utilize the formulas derived in section **3.1**. -The velocity of the particle will take the following form: +The velocity of the bodie will take the following form: $$ v' = \lim\limits_{m_w \to \infty} \frac{2m_wv_w + v(m-m_w)}{m+m_w} = \lim\limits_{m_w \to \infty} \frac{v(m-m_w)}{m+m_w} = -v @@ -114,7 +114,7 @@ $$ v_w' = \lim\limits_{m_w \to \infty} \frac{2mv + v_w(m_w-m)}{m+m_w} = \lim\limits_{m_w \to \infty} \frac{2mv}{m+m_w} = 0 $$ -This leads to the conclusion that the wall will not change its position, but the particle will change its velocity value to the opposite. +This leads to the conclusion that the wall will not change its position, but the bodie will change its velocity value to the opposite. ---