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I personally believe that after finding F = ma from Newton's second law we can write acceleration in two ways.

First Way a = dv/dt

F = m(dv/dt) ........ equation 1

Fdt = mdv

after integrating we get,

∫Fdt = mv - mu

Now, if we call product of mass with velocity as momentum and ∫Fdt as impulse then this will be our impulse-momentum theorem.

Also, if mv = p then from equation 1, we can say force is rate of change of momentum w.r.t. time.

I know this fact also that F = dp/dt is more generalized form of force and it can be applied in various fields of physics including electromagnetism and relativity, but I also know that laws cannot be proved but theorems can be proved. So, if we are trying to prove F = ma from F = dp/dt, is it a right thing to prove Newton's Law of Motion?

But the main problem is that I don't know which form of second law of motion is defined earlier (F = ma or F = dp/dt) and what is the exact second law of motion given by himself? and if I am thinking correctly in a right direction then who discovered impulse momentum theorem and did he / she find the same way in which I have found?

Second Way a = vdv/dx

F = m(vdv/dx)

Fdx = mvdv

after integrating we get,

∫Fdx = 0.5(mv^2) - 0.5(mu^2)

Now, if we call 0.5(mv^2) as kinetic energy and ∫Fdx as work then this will be our work-energy theorem.

But again, my doubt is that am I thinking correctly in a right direction then who discovered work-energy theorem, and did he / she find the same way in which I have found? Is this the same way in which work and kinetic energy are defined as first place or was there any other logic?

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