March 9, 2022

Momentum:

From Newton’s Second Law we know that $\vec{F}=m\vec a = m(d\vec v/dt)$. If $\vec F=0$, implying that $m\vec v$ is a constant. For many systems, $\vec F$ is 0. Therefore it becomes a sensible thing to study something which remains constant for such cases, namely $m\vec v$. This mathematical quantity, $m\vec v$ is called momentum and is denoted by $\vec p$. Thus, $\vec p = m\vec v$.

Consider a system of particles interacting with each other. Let each particle’s mass in the system be denoted by $m_i$, for a number of $n$ particles. Let the vector from the origin to that mass be $\vec {r}_i$. Therefore $\dot{\vec{r}}_i=\vec{v}_i$. Let the force on $i'th$ particle be $\vec {f}_i$. This force can come from two sources. First from particles included in the system and second from the particles outside the system. Therefore we can expand $\vec {f}_i$ as

$$ \vec {f_i} = {}^{int}\vec f_{i} + {}^{ext}\vec f_{i} \tag{3.1} \\ ‎ \\ \frac{d\vec p i}{dt}= {}^{int}\vec f{i} + {}^{ext}\vec f_{i} $$

If we add all the forces on all the particles then

$$ \sum \frac{d\vec p i}{dt} = \sum{}^{int}\vec f{i} + \sum{}^{ext}\vec f_{i} \tag{3.2} $$

From Newton’s third law we know that forces always come in pairs. Therefore $\sum{}^{int}\vec f_{i}=0$. Thus equation (3.2) reduces to

$$ \sum \frac{d\vec p i}{dt} = \sum{}^{ext}\vec f{i} \tag{3.3} $$

Writing the first term of equation (3.3) as $d\vec P /dt$ and the second term as $F$ we get

$$ \boxed{\frac {d\vec P}{dt}=\vec F} \tag{3.4} $$

Where $\vec P$ is the total momentum, since $\sum (dx/dt) = d\big(\sum x\big)/dt$ that is, the sum of derivatives is equal to the derivative of sum and $\vec F$ is the total external force. We can see that if $\vec F=0$, then $\vec P$ is a constant. The particles of a system may exchange momentum with each other by interacting and applying forces to each other, but the total never changes. That is, momentum is conserved for an isolated (free from external forces) system.

Center of mass:

Consider an inertial frame $S$ WRT in which some inertial frame $S'$ is moving with velocity $\vec u$. If a particle’s velocity in a system of $n$ particles is $\vec v_i$ WRT $S$ frame and $\vec v_i'$ WRT $S'$ frame then

$$ \vec v_i=\vec v_i'+\vec u \tag{3.5} $$