Question

derive a mathematical formulation of the second law of motion

Answer 1

EXPLANATION.

Let the mass of an object = m

initial velocity of an object = u

final velocity of an object = v

as we know that,

=> Momentum = mass X velocity

=> momentum of object at initial velocity u

=> m X u = mu

=> momentum of Object at final velocity v

=> m X v = mv

Change in momentum = mv - mumu

Rate of change in momentum =

=> mv - mu / t

According to the newton second law of

motion,

Force is directly proportional to the rate of

change of momentum.

$\rm \to \: force \: \propto \: rate \: of \: change \: of \: momentum$

$\rm \to \: force \: \propto \: \frac{mv \: - \: mu}{t}$

$\rm \to \: f \: \propto \: \frac{m(v \: - \: u)}{t}$

$\rm \to \: \therefore \: a \: = \frac{v \: - \: u}{t}$

$\rm \to \: f \: \propto \: ma$

$\rm \to \: f \: = kma$

=> k is known as proportionality constant.

$\green{\rm \to \therefore \: f \: = ma}$

Answer 2

Explanation:

Derivation of Newton's second law of motion

Newton's second law of motion states that " The rate of change of momentum is directly proportional to the force applied and takes place in the direction of applied force acts.

The mathematical expression:-

$\rm Force∝ \dfrac{Change \: in \: momentum}{Time \: taken}$

Rate of Change in momentum = mv - mu

Here:-

• Mass of the object = m

• Initial velocity = u

• Final velocity = v

•Time taken = t

• Force = F

So:-

$\rm \implies F∝ \dfrac{mv - mu}{t}$

$\rm \implies F∝ \dfrac{m(v - u)}{t}$

$\rm \implies F∝ m\dfrac{(v - u)}{t}$

As we know that

Acceleration is the rate of change of velocity.

$\rm\implies a = \dfrac{(v - u)}{t}$

$\rm \implies F∝ ma$

$\rm \implies F = k ma$

k = constant of proportionality and in SI units,it can be taken as 1

So:- k = 1

$\rm \implies F = (1) ma$

$\rm \implies F = ma$

The second law of motion

$\underline { \boxed{ \purple{\rm \therefore F = ma}}}$