Formula for Newton's First Law of Motion

Mathematical Formula For Newton’s First Law of Motion

It is very strange to note that, we have mathematical formula for Newton’s second law of motion F = ma, but no formula for Newton’s first law.

Simple Derivation of Newton’s Ist law From 2^nd Law

We know the relation between force and linear momentum

F = (P_f - P_i)/t_f -t_i

F = DP/Dt (Rate of change of linear momentum)

As an analogy

P = m(x_f - x_i)/t_f -t_i

P = mDx/Dt   

P = mv                          {v = Dx/Dt}

Momentum of a body is equal to the rate of change of change of position of a body of mass m which is nothing but Newton’s first law

If the position is not changing then its momentum is zero means the body is at rest P = 0

Usual Derivation of of Newton’s First Law

In the absence of an unbalanced force, linear momentum of a body is directly proportional to the its velocity and inversely proportional to its mass

(1) vµ P

(2) vµ 1/m

Combining (1) and (2), we have

vµ P/m

v = k P/m

when k =1 then

v = P/m

P = mv

Linear momentum is equal to the rate of change of position of a particle of mass m

Newton’s 2^nd Law of Motion from First Law

Writing momentum as the rate of change of posion of a body of mass m

DP = mDx/Dt

Dividing both sides by Dt

DP/Dt = mDx/DtDt

F = mDv/Dt

F = ma