Question

proof of conservation of momentum​

Answer 1

Law of Conservation of Linear Momentum

The law states that the total linear momentum of a system of particles remain constant unless an external unbalanced force is acting on it.

Proof based on Second Law:

By definition, external force is zero, i.e.,

$\displaystyle\longrightarrow\bf{F}\ \sf{=0}$

By Second Law of Motion,

$\displaystyle\longrightarrow\bf{F=\dfrac{\sf{d}\bf{p}}{\sf{dt}}}$

Thus,

$\displaystyle\longrightarrow\sf{\dfrac{d\bf{p}}{dt}=0}$

$\displaystyle\longrightarrow\sf{d\bf{p}\ \sf{=0}}$

By integration, we get,

$\displaystyle\longrightarrow\sf{\underline{\underline{\bf{p}\ \sf{=k,\ a\ constant}}}}$

Hence the Proof!

Proof based on Third Law:

Consider two masses $\displaystyle\sf{m_1}$ and $\displaystyle\sf{m_2}$ moving with velocities $\displaystyle\bf{u_1}$ and $\displaystyle\bf{u_2}$ respectively and they gain accelerations $\displaystyle\bf{a_1}$ and $\displaystyle\bf{a_2}$ after they collide each other.

Let $\displaystyle\sf{m_1}$ collide $\displaystyle\sf{m_2}$ by a force $\displaystyle\bf{F_1}$ so that the reaction exerted by $\displaystyle\sf{m_2}$ on $\displaystyle\sf{m_1}$ is taken as $\displaystyle\bf{F_2.}$

By Third Law of Motion,

$\displaystyle\longrightarrow\bf{F_1+F_2}\ \sf{=0}$

$\displaystyle\longrightarrow\sf{m_1\bf{a_1}}+\ \mathsf{m_2\bf{a_2}}\ \sf{=0}$

Let the impact be occurred for a time $\displaystyle\sf{t.}$ So,

$\displaystyle\longrightarrow\sf{m_1\bf{a_1}}\sf{t+m_2\bf{a_2}}\sf{t=0\quad\quad\dots(1)}$

But, by first kinematic equation,

$\displaystyle\longrightarrow\bf{v_1=u_1+a_1}\sf{t}\quad\implies\quad\bf{a_1}\sf{t}=\bf{v_1-u_1}$

$\displaystyle\longrightarrow\bf{v_2=u_2+a_2}\sf{t}\quad\implies\quad\bf{a_2}\sf{t}=\bf{v_2-u_2}$

where $\displaystyle\bf{v_1}$ and $\displaystyle\bf{v_2}$ are the velocities attained by $\displaystyle\sf{m_1}$ and $\displaystyle\sf{m_2}$ respectively after collision.

Then (1) becomes,

$\displaystyle\longrightarrow\sf{m_1(\bf{v_1-u_1})}+\sf{m_2(\bf{v_2-u_2})}=\sf{0}$

$\displaystyle\longrightarrow\sf{m_1\bf{v_1}-\sf{m_1}\bf{u_1}}+\sf{m_2\bf{v_2}-\sf{m_2}\bf{u_2}}=\sf{0}$

$\displaystyle\longrightarrow\sf{\underline{\underline{m_1\bf{u_1}+\sf{m_2}\bf{u_2}=\sf{m_1}\bf{v_1}+\sf{m_2}\bf{v_2}}}}$

This implies the total linear momentum of the system is conserved even after the collision.

Hence the Proof!