Question

derive mathematical formulation of Newton's second law ​

Answer 1

Answer:

Mathematical formulation of Newton's Second Law of Motion:

Let is initial velocity be u and final velocity be v. Now, 1 unit force is defined as the force applied on an object of mass 1kg to produce the acceleration of 1m/s2.

Answer 2

Explanation:

Let mass of an moving object = m. Let the velocity of the object changes from ‘u’ to ‘v’ in the interval of time ‘t’. This means,

Initial velocity of the object = u.

Final velocity of the object = v.

We know that momentum (p) = Mass x velocity

Therefore, Momentum (p) of the object at its initial velocity u = m x u = mu

Momentum (p) of the object at its final velocity v = m x v = mv

The change in momentum = mv – mu

Rate of change of momentum = mv - mu/t ...... (i)

According to Newton’s Second Law of motion force is directly proportional to the rate of change of momentum. This means, Force ∝ Rate of change of moentum After substituting the value of rate of change of momentum from equation (i) we get

$force \alpha \frac{mv - mu}{t}$

$f \alpha \frac{m(v - u)}{t}$

$f \alpha \: \: ma ....where \: a = \frac{v - u}{t}$

$f = k.ma...........(.ii)$

where, k is proportionality constant.

Since, 1 unit force is defined as the mass of 1kg object produces the acceleration of 1m/s2

Therefore, 1 unit of Force = k x 1 kg x 1m/s2

Thus k = 1.

By substituting the value of ‘k = 1’ in equation (ii) we get F = m.a ……………… (iii)

⇒ Force = mass × acceleration Thus Newton’s Second Law of Motion gives the relation between force, mass and acceleration of an object.

According to the relation obtained above, Newton’s Second Law can be modified as follows:

The product of mass and acceleration is the force acting on the object.

The SI unit of Force:

Newton (N) Since Force = Mass x Acceleration

The unit of mass = kg and The unit of acceleration = m/s2 If force, mass andacceleration is taken as 1 unit

. Therefore, 1 Newton (N) = 1kg x 1m/s2

Thus, Newton (N) = kg m/s2 Equation (iii) can be also written as a = F/m .....(iv)

This equation is the form of Newton’s Second Law of Motion.

Hope it helps you ❤️