# Derive the mathematical relation of Newton’s second law of motion.

Consider an object of mass m moving along a straight line with an initial velocity u (say). It is uniformly accelerated to velocity u in time t by the application of a constant force F in time t.

Then, initial momentum of the object = mu

P1 = mu

Final momentum of the object = mv

P2 = mv

∴ Change in momentum = mv mu = m(v u)

The rate of change in momentum = {𝑚 \times (v−u) \over 𝑡}

According to Newton’s second law of motion, we have

𝐹 ∞ {𝑚(𝑣−𝑢) \over 𝑡}

𝐹 = 𝑘𝑚 {(𝑣−𝑢) \over 𝑡}

F = k m a            …….(1)

Here, a = {𝑣−𝑢 \over t}= the rate of change of velocity.

= acceleration

k = a constant of proportionality

Putting m = 1 kg, a = 1 ms-2

F becomes 1 N.

So,   1 N = k x 1 kg x 1 ms-2

∴     k = 1

From equation (1), we have

F = ma

This represents the second law of motion.

Thus, the second law of motion gives a method to measure the force acting on an object as a product of its mass and acceleration.

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