Question

state the law of conservation of momentum and derive it from Newton's laws of motion ​

Answer 1

When force is applied on a moving body, its velocity changes. Due to the change in the velocity of the body, its momentum also changes.

Let a force $F$ be applied on a body of mass $m$ for time $t$ due to which its velocity changes from $u$ to $v$. Then

Initial momentum = mu

Final momentum = mv

Change in momentum of the body in $t$ second

$\implies\rm mv-mu=m(v-u)$

Rate of change of momentum

$\implies\rm \dfrac{Change \ in \ momentum}{Time} = \dfrac{m(v-u)}{t}$

But acceleration $a$ = $\rm \dfrac{Change \ in \ velocity}{Time}= \dfrac{v-u}{t}$

$\therefore \rm \ Rate \ of \ change \ of \ momentum = ma \qquad ......[1]$

On the basis of his experiments, Newton concluded that,

• The acceleration produced in a body of a given mass is directly proportional to the force applied on it.

$\rm a \propto F \ (if \ mass \ remains \ constant) \qquad ...[2]$

• The force needed to produce a given acceleration in a body is directly proportional to the mass of the body. i.e.,

$\rm F \propto m \ (if \ acceleration\ remains \ the \ same)\qquad ...[3]$

Combining equations [2] and [3],

$\boxed{\rm F \propto m \times a} \ or \ \boxed{F = Kma}\qquad ..[4]$

Here $K$ is constant. The unit of force is so chosen that $K$ becomes 1, when $m=1 \ and \ a=1$.

Thus,

$\boxed{\rm F = m\times a }\qquad ...[5]$

or force = mass × acceleration.

When force is applied on a body, it produces acceleration in the body due to which the velocity and hence the momentum of the body changes .

From equations [1], the rate of change of momentum is equal to the product of mass and acceleration i.e.,

$\qquad\qquad\qquad \rm\dfrac{\Delta p}{\Delta t}=ma ( if \ mass \ remains \ constant).$

From equation [5] by Newton's second law of motion, Force = ma

$\therefore \rm Force= Rate \ of \ change \ of \ momentum$

or $\rm F = \dfrac{\Delta p}{\Delta t}= \dfrac{\Delta (mv)}{\Delta t}= \dfrac{m \Delta t}{\Delta t}$

$\implies\rm ma ...(if \ mass \ remains \ constant)$

Thus Newton's second law of motion can be stated in terms of change in momentum as follows.

Statement.

According to Newton's second law of motion, the rate of change of momentum of a body is directly proportional to the force applied on it and the change in momentum takes place in the direction in which the force is applied.