Question

Write the law of conservation of momentum and prove it. Derive Newton's law of motion with the help of above law.

Answer 1

Law of Conservation of Linear Momentum:

According to Newton’s second law of motion the force acting on an isolated force is equal to rate of change of momentum i.e.,

\(\vec F = \frac{d\vec p}{dt}\) 

If external force is absent i.e. \(\vec F\) = 0

\(\frac{d \vec p}{dt} = 0\) 

or \(\vec p\) = constant (Because differentiation of constant with respect to any quantity is zero)

or p = mu = constant

or m₁u₁ = m₂u₂

i.e., in absence of external force the linear momentum of particle remains unchanged.

Proof of Newton’s Third Law by Newton’s Second Law:

Suppose there are two bodies A and B in an isolated system. When they collide, then at the time of collision, body A applies force \(\vec F_{21}\) on body Band body B applies force \(\vec F_{21}\) on body A These forces \(\vec F_{21}\) and \(\vec F _{12}\) are said action and reaction respectively. The time of contact of both particles is ∆t.

∵ Change in linear momentum = Force × time interval

∴ Change in momentum of particle A

\(\overrightarrow{\Delta p_1} = \overrightarrow{F_{12}}\times\Delta t\)

and that of particle B,

\(\overrightarrow{\Delta p_2} = \overrightarrow{F_{21}}\times\Delta t\)

∴ Total change in momentum of the system = \(\overrightarrow{\Delta p_1} + \overrightarrow{\Delta p_2}.\) If no external force is acting on the system, then according to law of conservation of momentum,

This is Newton's third law of motion.