Quantitative mechanical energy calculations combine kinetic energy, gravitational potential energy, and thermal energy dissipation. Applying exact energy formulas and Sankey efficiency diagrams enables complete system analysis in Cambridge IGCSE Physics.

1. Kinetic Energy ($E_k$)

Kinetic energy is the energy possessed by a body due to its motion. The kinetic energy of a body of mass $m$ moving at velocity $v$ is given by:

Because $v$ is squared, doubling an object's speed quadruples its kinetic energy ($2^2 = 4$).

2. Change in Gravitational Potential Energy ($\Delta E_p$)

Gravitational potential energy is the energy stored in a mass due to its position in a gravitational field. Raising a mass $m$ through vertical height $\Delta h$ in a uniform field $g$ increases its potential energy store by:

In an ideal conservative system with zero friction or air resistance, mechanical energy shifts continuously between $E_p$ and $E_k$ while total mechanical energy remains constant:

For an object dropped from rest ($u = 0$) at height $h$:

Notice that the final impact velocity $v$ is independent of the object's mass $m$!

In real physical systems, friction and air resistance dissipate mechanical energy into wasted thermal stores.

Efficiency Equations

Sankey diagrams represent energy paths visually: input energy enters from the left, useful energy flows straight ahead, and wasted energy branches off vertically. The width of each branch is strictly proportional to the energy quantity.

Roller coaster design relies on mechanical energy conservation. Motorized chains lift coaster cars to the top of the highest hill, storing maximum $\Delta E_p = mgh$. As the cars descend, potential energy converts into kinetic energy ($E_k = \frac{1}{2}mv^2$), yielding maximum speed at the lowest track trough. Subsequent loop heights must be strictly lower than the initial hill to compensate for friction losses.

Kinetic energy ($E_k = \frac{1}{2}mv^2$) and GPE ($\Delta E_p = mg\Delta h$) quantify mechanical state changes. Sankey diagrams and efficiency formulas evaluate energy transformations in mechanical and industrial systems.