Energy

• What energy is and why it is measured in Joules
• Kinetic and potential energy formulas with worked derivations
• The law of conservation of energy
• How energy flows from the Sun through photosynthesis to your body

What is energy? Energy is the capacity (ability) to do work. It is a scalar quantity measured in Joules (J). An object or system has energy if it can exert a force over a distance — i.e., if it can do work.

Energy comes in many forms: kinetic, potential, chemical, thermal, electrical, nuclear, sound, light. A key principle: energy cannot be created or destroyed, only transformed from one form to another. This is the Law of Conservation of Energy.

Kinetic Energy (KE) Kinetic energy is the energy an object possesses due to its motion:

KE = (1/2) × m × v²

where m is mass (kg) and v is speed (m/s). KE is always positive (or zero). It increases with the square of speed — doubling speed quadruples KE.

Potential Energy (PE) Gravitational potential energy is the energy stored in an object due to its height above a reference level:

PE = m × g × h

where m is mass (kg), g is gravitational acceleration (9.8 m/s²), and h is height (m). Potential energy depends on the choice of reference level (usually the ground), but changes in PE do not depend on reference.

Conservation of Energy In the absence of non-conservative forces (like friction), total mechanical energy is conserved:

KE + PE = constant

This means: as an object falls, PE decreases and KE increases by exactly the same amount. At any height h (measuring from the ground):

(1/2)mv² + mgh = constant = mgh_initial (if starting from rest)

At the bottom (h = 0): all energy is kinetic, so (1/2)mv² = mgh_initial → v = √(2gh)

Energy chain: Sun to rice bowl

1. Nuclear fusion in the Sun releases vast energy as electromagnetic radiation (light and heat).
2. Light travels 150 million km to Earth in about 8 minutes.
3. Rice plants absorb sunlight in chlorophyll via photosynthesis: 6CO₂ + 6H₂O + light energy → C₆H₁₂O₆ (glucose) + 6O₂ Light energy is stored as chemical energy in the glucose molecules.
4. The rice grain stores this chemical energy as starch (chains of glucose).
5. When you eat rice, your digestive system breaks starch back into glucose.
6. Your cells perform cellular respiration to release this chemical energy.
7. The energy powers muscle contractions (mechanical energy), body heat (thermal energy), and nerve signals (electrical energy).

Every calorie of food energy traces back to sunlight.

Forms of energy — brief reference

• Chemical: stored in molecular bonds (food, fuel, batteries)
• Thermal: due to random motion of particles (heat)
• Electrical: due to flow of charges
• Nuclear: stored in atomic nuclei (released in fission/fusion)
• Sound: mechanical vibrations through a medium

Problem: A 2 kg ball is held at a height of 5 m above the ground and then released from rest. (a) What is the potential energy at the top? (b) What is the kinetic energy just before hitting the ground? (c) What is the speed just before impact?

Solution:

(a) PE at the top: PE = mgh = 2 × 10 × 5 = 100 J (using g = 10 m/s² for simplicity)

(b) By conservation of energy (no friction/air resistance): KE at bottom = PE at top = 100 J

(c) KE = (1/2)mv² 100 = (1/2)(2)v² 100 = v² v = 10 m/s

Check using kinematics: v² = 2gh = 2 × 10 × 5 = 100, so v = 10 m/s ✓

• Thinking energy is "used up." Energy is never consumed — it is transformed. When a light bulb is "using electricity," it converts electrical energy to light and heat. Total energy before = total energy after.
• Forgetting that KE depends on v², not v. A car at 60 km/h has four times the KE of the same car at 30 km/h, making high-speed crashes far more dangerous.
• Using PE = mgh at the wrong reference. Clearly define where h = 0. PE can be calculated as the change in height from that reference; only differences in PE matter for energy conservation.
• Applying conservation of mechanical energy when friction is present. Friction converts mechanical energy to thermal energy. In that case, KE + PE at start = KE + PE at end + heat generated by friction.

1. A 5 kg object moving at 6 m/s has what kinetic energy?
2. A 3 kg book is placed on a shelf 2 m above the floor. What is its gravitational PE? (g = 10 m/s²)
3. If a ball rolls off a table of height 0.8 m, what is its speed when it hits the ground? (g = 10 m/s², ignore air resistance)