Show the relationship between momentum and force OR Derive Newton’s Second Law of motion with the help of momentum.

OR

Prove that $F=\frac{\triangle P}{t}$

OR

How can you relate a force with the change of momentum of a body?

Difficulty: Medium

Force and momentum:

Consider a body of mass m moving with initial velocity vi, let a force acts on the body which produces an acceleration an in it. This changes the velocity of the body. Let its final velocity after time t become vf. If pi and pf be the initial momentum and final momentum of the body related to initial and final velocities respectively then

Pi=mvi

Pf=mvf

Or

Change in momentum = final momentum – initial momentum

Or

P=mv

Thus the rate of change in momentum is given by:

$\frac{pf \:, \: pi}{t} =\frac{mvf \: - mvi}{t}$

Since $\frac{vf \:- \: vi}{t}$ is the rate of change of velocity equal to the acceleration a produced by the force F

$\frac{pf \:, \: pi}{t}$ = ma

According to Newton’s second law of motion,

F= ma

Or $\frac{pf \:, \: pi}{t}$ = F …………… (I)

Equation (I) also defines force and states Newton’s second law of motion as:

When a force acts on a body, it produces an acceleration in the body and will be equal to the rate of change of momentum of the body.