Question

What is Newton’s second law of motion and do we derive the formula for force from this?

Answer 1

Hint: In order to solve this question, first we need to understand what Newton's laws of motion actually are. Newton's laws of motion are basically the three laws of motion that are used to describe the relationship between the motion of an object and the forces acting on it.

Complete step by step answer:
According to Newton’s second law of motion, the rate of change of momentum of a body is directly proportional to the force applied on it. Let, $\overrightarrow p $ be the momentum of the body, $m$ be the mass of the body and $\overrightarrow v $ be the velocity of the body.Then,
$\overrightarrow p = m\overrightarrow v $
According to the Newton’s second equation of motion,
$F \propto \dfrac{{d\overrightarrow p }}{{dt}}$
On putting the value of $\overrightarrow p $ in the above equation,
$F \propto \dfrac{d}{{dt}}(m\overrightarrow v )$
Since, $m$ is a constant, so,
$F \propto m\dfrac{d}{{dt}}(\overrightarrow v )$
As $a = \dfrac{{d\overrightarrow v }}{{dt}}$, so the above expression can be written as,
$F \propto ma$
On replacing the proportionality sign with a constant, we get,
\[F = kma\]
If $k = 1$, then the above equation becomes,
$F = ma$
Where, $F$ is the force applied, $m$ is the mass of the body and $a$ is the acceleration of the body.
The above equation is the mathematical representation of Newton's second law of motion.

Note: While deriving the expression for force$F = ma$, we consider mass to be constant in this case. We also consider the proportionality constant to be unity. When we do relativistic study of the motion of an object, i.e., where the speed of the object becomes equal to the speed of the light, then the formula $F = ma$ is not applicable. The reason for this is that the mass of the object becomes variable in relativistic study.