An object at a high position has the ability to do work by falling. This is potential energy.
Basic Knowledge
Gravitational potential energy U:U = mgh
m: mass(kg), g: gravitational acceleration(m/s²), h: height(m) above the reference level. The unit is J.
As height increases, energy increases. The reference level(the level defined as h=0) can be set freely.
Mechanical energy = kinetic energy + potential energy:E = K + U = ½mv² + mgh
Law of conservation of mechanical energy: When external forces(friction, air resistance, etc.) do no work, mechanical energy remains constant:½mv₁² + mgh₁ = ½mv₂² + mgh₂
This formula expresses mathematically that “as height decreases, speed increases, and as height increases, speed decreases.”
Potential energy U (potential energy)（mgh. The ability to do work according to height）
Mechanical energy E（The total of K + U. Conserved without friction）
Law of conservation of mechanical energy（E = constant when external forces do no work）
Reference level（The level set as h = 0. It can be chosen freely）
Potential energy due to elastic force（Spring: U = ½kx². Spring constant k, extension x）
Energy conversion（Conversion between potential energy ↔ kinetic energy）
Further Study (Background and Significance)
The law of conservation of mechanical energy holds only when there is no friction or air resistance. Since these are always present in real motion, mechanical energy decreases and is converted into heat and other forms of energy.
Roller coasters are a typical example of conservation of mechanical energy. Potential energy at the initial high point is converted into kinetic energy during descent, making the coaster move quickly. Because the energy conversion is carefully designed, the speed at the next point can be calculated from the starting height(if energy is perfectly conserved).
Potential energy due to elastic force(the spring energy U = ½kx²) is also conserved in the same way. In a spring-mass system, U and K convert alternately while the system oscillates(simple harmonic motion).
This conservation law is the mechanical version of the law of conservation of energy(the first law of thermodynamics), which states that “energy does not disappear.”
- U = mgh(proportional to height above the reference level)
- E = K + U = ½mv² + mgh
- Conservation law: no friction → E = constant
- Height↑ → U↑ → K↓ → v↓
- Height↓ → U↓ → K↑ → v↑
- The reference level can be chosen freely
- When friction is present, E decreases
Points to Note (Common Confusions)
① In U = mgh, h is the height above the reference level. Since the reference level differs between problems, state it clearly. ② The conservation law can be used only when there is no friction or air resistance. When friction is present, calculate using “lost mechanical energy = frictional force × distance traveled.” ③ Changing the reference level does not change the result of conservation-law calculations(because differences are taken). ④ A point where speed becomes 0 is not necessarily the highest point(there may still be horizontal velocity).
Practice
- An object of mass 2 kg is at a height of 10 m. Taking the ground as the reference level, find its potential energy(g = 10 m/s²).
- When the object above undergoes free fall, find its speed at a height of 5 m using the law of conservation of mechanical energy.
- On a frictionless course, an object of mass 3 kg starts from rest at a height of 20 m. Find its speed when it reaches the ground(g = 10 m/s²).