Energy Conservation — Algebra Keeps the Universe in Balance
Conservation of Energy
One of the most powerful ideas in all of physics can be expressed as a single algebraic equation:
$$KE_1 + PE_1 = KE_2 + PE_2$$
$$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$$
where $m$ is mass, $v$ is velocity, $g = 9.8$ m/s², and $h$ is height.
This is an algebraic equation — plug in what you know, solve for what you don't. The mass $m$ often cancels out, making it even simpler.
Roller Coasters — Energy in Action
At the top of a roller coaster hill (height $h$, speed ≈ 0), all energy is potential. At the bottom, all energy is kinetic:
$$mgh = \frac{1}{2}mv^2 \quad \Rightarrow \quad v = \sqrt{2gh}$$
Notice: the mass cancels! A heavy car and a light car reach the same speed at the bottom. This is why Galileo said all objects fall at the same rate.
A roller coaster starts from rest at the top of a 40-meter hill. How fast is it moving at the bottom (ignoring friction)?
$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 40} = \sqrt{784} = 28.0 \text{ m/s}$$
That's 100.8 km/h — from a simple square root!
A pendulum is 2 meters long and released from 30° to the vertical. What is its speed at the lowest point?
Height gained: $h = L(1 - \cos\theta) = 2(1 - \cos 30°) = 2(1 - 0.866) = 0.268$ m
$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 0.268} = \sqrt{5.25} \approx 2.29 \text{ m/s}$$
Trigonometry ($\cos\theta$) and the square root combine to predict the swing.
Energy conservation is an algebraic equation with squares, square roots, and sometimes trig. It lets you solve problems that would require complex calculus if done any other way. It's one of the most useful equations in all of physics.