Conservation of Energy
As an object moves, its energy can shift back and forth between motion (kinetic energy) and position (potential energy), but the total mechanical energy stays the same as long as friction and air resistance aren't stealing any away. A falling object trades height for speed, and a rising object trades speed for height — the sum never changes.
The formula
E = ½ · m · v² + m · g · h
- E — total mechanical energy (J): the combined energy of motion and position, constant without friction
- m — mass (kg): how much matter the object has
- v — speed (m/s): how fast the object is moving at that moment
- g — gravitational acceleration (m/s²): the strength of gravity's pull, about 9.8 m/s² near Earth's surface
- h — height (m): how far above a reference level the object is
Worked example
A 2 kg ball swings through its path at a height of 5 m above the ground, moving at 4 m/s. Using g = 9.8 meters per second squared, what is its total mechanical energy?
- m = 2 kg
- v = 4 m/s
- h = 5 m
- E = ½ · m · v² + m · g · h
- E = ½ · 2 kg · (4 m/s)² + 2 kg · 9.8 m/s² · 5 m
E = 114 J
Test yourself
The same 2 kg ball now reaches a height of 10 m, still moving at 4 m/s at that point. What is its total mechanical energy?
- 228 J
- Correct answer: 212 J
- 114 J
Right! E = ½ × 2 kg × (4 m/s)² + 2 kg × 9.8 m/s² × 10 m = 16 J + 196 J = 212 J.
A ball rolls down a frictionless hill, losing height as it goes. What happens to its speed and its total mechanical energy?
- Correct answer: Speed increases, total energy stays the same
- Speed decreases, total energy stays the same
- Speed increases, total energy increases
Exactly! Height turns into speed — kinetic energy goes up as potential energy goes down, but the total stays constant without friction.
Where you see this
A pendulum swings highest at the ends, where it's briefly still, and fastest at the bottom, trading height for speed and back again every swing. A roller coaster's tallest hill is always its first — the ride can never rise higher than its starting point once it starts converting height into speed.
Common mistakes
A frequent error is assuming energy "runs out" when something slows down — it doesn't disappear, it converts into another form, usually heat from friction or air resistance that a simple mechanical-energy total doesn't track. Another is mixing up kinetic and potential energy's dependence: potential energy (mgh) depends only on height, not speed, while kinetic energy (½mv²) depends only on speed, not height.
How it connects
This is the second half of the toolkit conservation of momentum started: momentum tracks direction and always conserves in a closed system, while mechanical energy tracks motion-plus-position and only conserves when friction isn't stealing any away. Torque and levers, next, apply the same force ideas to rotation instead of straight-line motion.