Introduction
The Principle of Conservation of Energy is a cornerstone of classical mechanics. It states that energy cannot be created or destroyed, only transformed from one form to another. In an isolated system, the total energy remains constant.
Total Mechanical Energy
Total mechanical energy ($E$) is defined as the sum of kinetic energy ($K$) and potential energy ($U$): $$E = K + U$$
Proof: Energy Conservation in a Gravitational Field
Consider a body of mass $m$ falling freely from a height $h$ above the ground. Let's analyze the energy at three points: the starting point ($A$), a point during the fall ($B$), and the ground ($C$).
1. At point A (Initial height $h$)
- Kinetic Energy ($K_A$): $0$ (starting from rest)
- Potential Energy ($U_A$): $mgh$
- Total Energy ($E_A$): $0 + mgh = mgh$
2. At point B (Distance $x$ below A)
- The distance from the ground is now $h-x$. The velocity $v$ after falling $x$ can be found using $v^2 = u^2 + 2ax$. Here $u=0$ and $a=g$, so $v^2 = 2gx$.
- Kinetic Energy ($K_B$): $\frac{1}{2}mv^2 = \frac{1}{2}m(2gx) = mgx$
- Potential Energy ($U_B$): $mg(h-x) = mgh - mgx$
- Total Energy ($E_B$): $K_B + U_B = mgx + mgh - mgx = mgh$
3. At point C (At the ground)
- The body has fallen the full height $h$. $v^2 = 2gh$.
- Kinetic Energy ($K_C$): $\frac{1}{2}m(2gh) = mgh$
- Potential Energy ($U_C$): $0$ (height is $0$)
- Total Energy ($E_C$): $mgh + 0 = mgh$
Conclusion
Since $E_A = E_B = E_C = mgh$, we have demonstrated that the total mechanical energy is conserved under the action of a gravitational field.