Question

Verify the principle of conservation of linear momentum using Newton’s second law of motion only?

Answer 1

Hint: In classical mechanics, Newton’s laws of motion are three laws that describe the relationship between the motion of an object and the forces acting on it. The second law states that the rate of change of momentum of an object is directly proportional to the force applied or the net force on an object is equal to the mass of that object multiplied by its acceleration.

Complete step by step answer:
As described in the hint section, according to Newton’s second law, force is the multiplication of the mass of the object and its acceleration.
${\text{F = ma}}\,\,\,\,\,.....{\text{(i)}}$
Where, $F$ is the force applied, ${\text{m}}$ is the mass of the object and ${\text{a}}$ is the acceleration of that object.
Now, we can write acceleration as
$${\text{a = }}\dfrac{{\Delta {\text{v}}}}{{\Delta t}}\,\,\,\,\,......({\text{ii}})$$
Here in the above equation $$\Delta {\text{v}}$$ is the change in velocity of the object over some time duration $$\Delta t$$ .
Now from $({\text{i}})$ and $({\text{ii}})$, we can write
${\text{F = m}}\dfrac{{\Delta {\text{v}}}}{{\Delta t}}\,\,\,\,\,.....{\text{(iii)}}$
As equation of change in momentum is
$\Delta {\text{p = m}}\Delta {\text{v }}....{\text{(iv)}}$
From $({\text{iii}})$ and $({\text{iv}})$ , we can write
${\text{F = }}\dfrac{{\Delta {\text{p}}}}{{\Delta t}}$
We know that objects apply force on each other, they are equal and opposite.
Therefore, we can write
${{\text{F}}_{{\text{a on b}}}} = - {{\text{F}}_{{\text{b on a}}}}$
$ \Rightarrow \dfrac{{\Delta {{\text{p}}_{\text{a}}}}}{{\Delta t}} = - \dfrac{{\Delta {{\text{p}}_{\text{b}}}}}{{\Delta t}}$
Which can be write as
$ \Rightarrow \Delta {{\text{p}}_{\text{a}}} = - \Delta {{\text{p}}_{\text{b}}}$
$ \Rightarrow \Delta {{\text{p}}_{\text{a}}} + \Delta {{\text{p}}_{\text{b}}} = 0$
So change in momentum of an object a and b are equal and opposite.
This demonstrates conservation of momentum, since the sum of the momentum of the object is always zero.

Note: According to the law of conservation of momentum, when two or more bodies act upon one another, their total momentum remains constant provided no external forces are acting. We can also say that momentum can never be created nor be destroyed.