Question

law of conservation of energy derivation

Answer 1

Hi there,
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time. This law means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
hope i am right

Answer 2

It is a direct consequence of Newton’s Second Law:

(1) F=ma=md2xdt2F=ma=md2xdt2

where I am using boldface to represent vector quantities, and x=x(t)x=x(t) is the vector of time-dependent coordinates. In non-relativistic mechanics, the energy is defined as

E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)E=12mv2+V(x)=12mdxdt⋅dxdt+V(x)

where V(x)V(x) is the potential energy. The force is the (negative) spatial derivative of the potential energy,

(2) F=−∂∂xV(x)=−∇V(x)F=−∂∂xV(x)=−∇V(x)

If we take the total time derivative of the energy,

dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))dEdt=12mddtdx(t)dt⋅dx(t)dt+ddtV(x(t))

and apply the product rule and the chain rule,

dEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdtdEdt=mdxdt⋅d2xdt2+∂∂xV(x(t))⋅dxdt

But, using (1) and (2), we find

dEdt=dxdt⋅F−F⋅dxdt=0dEdt=dxdt⋅F−F⋅dxdt=0

This says that the rate of change of the energy is always zero. Hence, energy is a conserved quantity.