8. Mathematical formulation of second law

Consider an object of mass, \(m\) with an initial velocity, \(u\) is moving along a straight line. It is in uniform acceleration with velocity, \(v\) in time, \(t\) by a constant force, \(F\).

 

Initial momentum of the object, \(p_1\) \(=\) \(m\ u\)
Final momentum of the object, \(p_2\) \(=\) \(m\ v\)

 

 

 

Where \(k\) is the constant of proportionality.

 

 

Here, $a=\frac{\left(v-u\right)}{t}$

 

The above equation, which implies the rate of change of velocity, is known as acceleration.

We know that the SI units of mass is \(kg\) and acceleration is \(m\ s^{ –2}\). The unit of force is selected in such a way that the value of the proportionality constant, \(k\) is one.

 

Hence, the value of \(k\) becomes one and makes $F=ma$.

 

The SI unit of force is \(kg\ m\ s^{ –2}\) or newton (\(N\)). The value of one newton is written as,

\(1\ N\) \(=\) \(1\ kg\ m\ s^{ –2}\)