Question

Derive the mathematical formulation of Newton’s second law of motion $ F = ma $ .

Answer 1

Hint
The mathematical formulation of newton’s second law can be derived with the help of the concept of momentum. The second law of motion defines the force as a quantity which is the rate of change of momentum. Also, acceleration can be defined as the rate of change of velocity, and momentum is defined as mass times velocity.

Complete step by step answer
Newton’s second law states that the rate of change of momentum of a body is directly proportional to the applied force and this change occurs in the same direction in which the force acts.
For the mathematical derivation, we assume a moving point object of mass, m.
Let its initial velocity be u.
Let its final velocity be v.
The momentum of this body can be given by, $ p = mass \times velocity $
Therefore, the initial momentum of the body is, $ {p_i} = mu $
The final momentum of the body is, $ {p_f} = mv $
The change in momentum can be given by,
 $\Rightarrow \vartriangle p = {p_f} - {p_i} $
 $\Rightarrow \vartriangle p = mv - mu $
Let the time interval where this change of momentum takes place be $ t $ . Then,
Rate of change of momentum is given by,
 $\Rightarrow \dfrac{{\vartriangle p}}{t} = \dfrac{{mv - mu}}{t} $
The expression $ \dfrac{{mv - mu}}{t} $ can be rewritten as $ \dfrac{{m(v - u)}}{t} $
And it is known that $ \dfrac{{v - u}}{t} $ is the expression for acceleration ( $ a $ ) of the body.
Substituting this value,
 $\Rightarrow \dfrac{{\vartriangle p}}{t} = m\left( {\dfrac{{v - u}}{t}} \right) $
From Newton’s second law of motion, we know that,
 $\Rightarrow F \propto \dfrac{{\vartriangle p}}{t} $
So, $ F \propto \dfrac{{mv - mu}}{t} $
 $\Rightarrow F \propto ma $
On removing the proportionality sign,
 $\Rightarrow F = kma $
Where k is the proportionality constant.
To find the value of this proportionality constant, the definition of a unit force is taken into account,
1 unit force can be defined as the force applied on a 1kg object that would produce an acceleration of $ 1m/{s^2} $ .
On equating this,
1 unit of force $ = k \times \left( {1kg} \right) \times \left( {1m/{s^2}} \right) $
Therefore $ k = \dfrac{1}{{1 \times 1}} = 1 $
Therefore the value of k is taken 1.
Now the equation becomes,
 $\Rightarrow F = ma $

Note
Force is a vector quantity. It has the same direction as the acceleration. The law of conservation of momentum is also derived from Newton’s second law of motion, it says that the net momentum of a system remains constant unless an external force acts upon it.