Law of Conservation of Energy

The law of conservation of energy states that energy can neither be created nor destroyed. It can only be transformed from one form into another or transferred from one object to another, but the total amount of energy in an isolated system never changes. This is one of the most fundamental principles in all of physics, and it applies to everything from a swinging pendulum to a car engine to the chemical reactions inside your body.

In symbols, if we add up every form of energy in a closed system before and after some event, the totals must match:

\(E_{total,\,initial} = E_{total,\,final}\)

This idea connects directly to the broader topic of work and energy, since doing work on a system is really just transferring energy into or out of it.

In many introductory problems, we only need to track mechanical energy, which is the sum of kinetic energy (the energy of motion) and potential energy (stored energy due to position, usually height). The conservation of mechanical energy formula is:

\(KE_i + PE_i = KE_f + PE_f\)

where kinetic energy and gravitational potential energy are calculated as:

\(KE = \frac{1}{2}mv^2\)    and    \(PE = mgh\)

Here \(m\) is mass, \(v\) is speed, \(g\) is the acceleration due to gravity, and \(h\) is height above a chosen reference point. This version of the law only holds true when no energy is lost to friction, air resistance, or sound. When those forces are present, some mechanical energy converts into thermal energy, but the total energy of the whole system, mechanical plus thermal plus any other forms, is still conserved.

A swinging pendulum is a classic way to see the law of conservation of energy in action. At the highest point of the swing, the pendulum momentarily stops, so all of its mechanical energy is potential energy. As it swings down, that potential energy converts into kinetic energy, reaching maximum speed (and maximum kinetic energy) at the lowest point. As it rises again on the other side, kinetic energy converts back into potential energy.

At every single point along the swing, if you add the kinetic energy and potential energy together, you get the same total value, ignoring the small amount lost to air resistance. This is exactly what the conservation of mechanical energy formula predicts.

A \(2\ \)kg\(\) ball is dropped from a height of \(5\ \)m\(\). Find its speed just before it hits the ground, assuming air resistance is negligible and \(g = 9.8\ \)m/s\(^2\).

At the top, the ball is at rest, so all of its energy is potential: \(PE_i = mgh = (2)(9.8)(5) = 98\ \)J\(\), and \(KE_i = 0\).

Just before hitting the ground, its height is \(0\), so \(PE_f = 0\), and all of the energy has converted into kinetic energy: \(KE_f = 98\ \)J\(\).

Using \(KE_f = \frac{1}{2}mv^2\), solve for \(v\):

\(98 = \frac{1}{2}(2)v^2 \Rightarrow v^2 = 98 \Rightarrow v \approx 9.9\ \)m/s\(\)

Notice that the mass canceled out of the final answer, which is a common feature of free-fall problems solved using conservation of mechanical energy.

Students sometimes worry that energy "disappears" when a ball stops bouncing or a swing slows down. It doesn't disappear, it transforms into forms that are harder to track, usually heat from friction or air resistance, and sometimes sound. If you could measure every joule of thermal energy released, the books would balance exactly, confirming that energy is neither created nor destroyed, only converted. This distinction between energy conservation and energy efficiency is explored further in power and efficiency, which looks at how usefully energy is transferred rather than whether it is conserved.

The law of conservation of energy is a universal principle: the total energy of an isolated system stays constant over time. For mechanical systems free of friction, this simplifies to the conservation of mechanical energy formula, \(KE_i + PE_i = KE_f + PE_f\), which lets you solve for an unknown speed, height, or mass without needing to know anything about the path taken between the initial and final states.

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