Mechanical Energy
Mechanical energy is the sum of an object's kinetic and potential energy (ME = KE + PE), conserved when only gravity or springs act.
Formula
ME = KE + PE
Definition
Mechanical energy is the energy of motion plus the energy of position added together: the total energy an object has because of how it moves and where it is. It is the sum of kinetic and potential energy, $ME = KE + PE$, and when only gravity or elastic forces do work, with no friction or air resistance, it stays constant even as kinetic and potential energy trade back and forth; friction converts mechanical energy into thermal energy. In general, $E = K + U$ is conserved whenever all forces doing work are conservative. With non-conservative forces present, the work-energy theorem generalizes to $W_{\text{nc}} = \Delta K + \Delta U = \Delta E$, so friction and drag reduce mechanical energy by exactly the thermal energy they generate. Conservation of $E$ is a first integral of Newton's second law and often replaces solving the equation of motion.
Example
A swing at the top of its arc has stored energy, and at the bottom it has moving energy; add the two together at any moment and you get the swing's mechanical energy. A $2\ \text{kg}$ ball at $4\ \text{m}$ height moving at $3\ \text{m/s}$ has $PE = 2 \times 9.8 \times 4 = 78.4\ \text{J}$ and $KE = 0.5 \times 2 \times 9 = 9\ \text{J}$, so $ME = 87.4\ \text{J}$, and at the ground where $PE = 0$ all $87.4\ \text{J}$ is kinetic. With friction present, a $50\ \text{kg}$ sled that starts from rest at the top of a $12\ \text{m}$ hill and arrives at the bottom at $12\ \text{m/s}$ began with $E = 50 \times 9.8 \times 12 = 5{,}880\ \text{J}$ and ended with $K = 0.5 \times 50 \times 144 = 3{,}600\ \text{J}$. Friction removed $2{,}280\ \text{J}$ as heat along the way.
Key Insight
On a swing, energy sloshes back and forth between stored and moving, but the total stays the same until friction slowly steals it. A roller coaster has the same mechanical energy at the top of the first hill as at the bottom, ignoring friction: it is fast where it is low and slow where it is high, and the total never changes. This is the practical power of the idea. Energy conservation gives speeds without knowing the path, the shape of the hill, or the time taken; where Newton's laws demand the full trajectory, energy methods need only two snapshots.