Kinetic Energy (KE) = ½mv². A 70 kg person running at 4 m/s: KE = ½ × 70 × 16 = 560 Joules. Gravitational Potential Energy (GPE) = mgh. That same person at the top of a 3-meter diving board: GPE = 70 × 9.81 × 3 = 2,060 Joules.
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What Is Energy in Physics?
In physics, energy is the capacity to do work — to exert a force over a distance. It is a scalar quantity measured in Joules (J) in the SI system. Energy exists in many forms: kinetic (energy of motion), gravitational potential (energy stored by height), elastic potential (energy stored in compressed or stretched springs), thermal, chemical, electrical, and nuclear. A fundamental law of physics — the conservation of energy — states that energy cannot be created or destroyed, only converted from one form to another.
Mechanical energy, the focus of most introductory physics problems, combines kinetic and potential energy. A ball thrown into the air converts kinetic energy to gravitational potential energy as it rises, then back to kinetic energy as it falls. In an ideal system with no friction or air resistance, the total mechanical energy remains constant throughout.
Kinetic energy depends on mass and the square of velocity. This square relationship means that doubling speed quadruples kinetic energy — which is why vehicle collision speeds have such a dramatic effect on impact forces and why speed limits matter so much for safety.
Gravitational potential energy depends on mass, gravitational acceleration (9.81 m/s² at Earth's surface), and height above a reference point. It represents the work that gravity could do if the object were allowed to fall. These two forms of energy are central to problems in mechanics, engineering design, sports science, and everyday physics.
Energy Formulas
Kinetic Energy:
KE = ½mv²
Where: m = mass (kg), v = velocity (m/s)
KE units: Joules (J)
Gravitational Potential Energy:
GPE = mgh
Where: m = mass (kg), g = 9.81 m/s², h = height (meters)
Elastic Potential Energy (spring):
EPE = ½kx²
Where: k = spring constant (N/m), x = compression/extension (m)
Work:
W = F × d × cos(θ)
Where: F = force (N), d = distance (m), θ = angle between force and displacement
Conservation of Energy
In an ideal system with no friction or air resistance, total mechanical energy is conserved:
KE + GPE = constant
At bottom (all KE): ½mv² = mgh (at top, all GPE)
Therefore: v = √(2gh)
A ball dropped from 10 meters:
- At top: GPE = m × 9.81 × 10 = 98.1m J; KE = 0
- Just before impact: GPE = 0; KE = 98.1m J
- Impact velocity: v = √(2 × 9.81 × 10) = 14 m/s (31.3 mph)
Energy Unit Conversions
| Unit | Symbol | Equivalent | |------|--------|-----------| | Joule | J | Base SI unit | | Kilojoule | kJ | 1,000 J | | Calorie (food) | kcal | 4,184 J | | Watt-hour | Wh | 3,600 J | | Kilowatt-hour | kWh | 3,600,000 J | | BTU (British) | BTU | 1,055 J | | Foot-pound | ft·lb | 1.356 J | | Electronvolt | eV | 1.602 × 10⁻¹⁹ J |
Common Energy Calculations
Impact energy of a falling object: A 0.5 kg wrench dropped 5 meters: GPE = 0.5 × 9.81 × 5 = 24.5 J (all converted to KE on impact). Impact velocity = √(2×9.81×5) = 9.9 m/s (22 mph). This is why dropped tools from heights are deadly — even small objects at 20+ mph carry significant impact force.
Crash kinetic energy: A 1,500 kg car at 60 mph (26.8 m/s): KE = ½ × 1,500 × 26.8² = 539,000 J (539 kJ). At 30 mph (13.4 m/s): KE = ½ × 1,500 × 13.4² = 134,730 J — one quarter the energy, because KE scales with velocity squared.
Food energy to mechanical work: A 200-calorie banana = 200 kcal = 836,800 J. A 70 kg person burning 200 cal climbing stairs at 100% efficiency could climb: h = 836,800 / (70 × 9.81) = 1,220 meters (4,000 feet). In practice, the body is ~25% efficient for mechanical work, so actual climb = ~305 meters.
Frequently Asked Questions
What is the work-energy theorem? The work-energy theorem states that the net work done on an object equals the change in kinetic energy: W_net = ΔKE = KE_final − KE_initial. If a 10 N net force acts over 5 meters on a 2 kg object starting from rest: W = 10 × 5 = 50 J; KE gained = 50 J; Final velocity = √(2 × 50 / 2) = 7.07 m/s. This is one of the most useful theorems in mechanics because it relates force and motion without requiring detailed analysis of every intermediate instant.
What is the difference between energy and power? Energy is the total amount of work done or capacity to do work (Joules). Power is the rate of energy transfer or work done per unit time (Watts = Joules/second). A 100-watt light bulb uses 100 joules of electrical energy per second. Running up stairs in 10 seconds requires the same total energy as walking up in 60 seconds — but running requires 6× the power output. Your electricity bill charges for energy (kWh), not power (kW).
What are Joules in everyday terms? One Joule is roughly: the energy to lift a 100g apple (quarter-pound) one meter; the kinetic energy of a tennis ball moving at about 6 mph; 1/4 of a calorie. 1 food Calorie (kcal) = 4,184 J. A cheeseburger (~500 Calories) contains ~2,092,000 J of chemical energy — far more than the mechanical energy it enables you to exert because the body converts only ~25% of food energy to mechanical work.
How do roller coasters use the energy-conservation principle? A roller coaster is powered initially to the top of the first hill, giving it maximum gravitational potential energy. As it descends, PE converts to KE (speed). As it climbs the next hill, KE converts back to PE. Without friction and air resistance, the coaster could return to its starting height. In reality, energy is lost to friction and air resistance — subsequent hills must be lower than the first. Designers calculate the energy at every point to ensure sufficient speed through loops and inversions.
What is binding energy? Nuclear binding energy is the energy required to separate an atomic nucleus into its component protons and neutrons. Nuclear fission (splitting heavy atoms) and fusion (combining light atoms) release this binding energy. Fission of 1 kg of uranium-235 releases approximately 6.7 × 10¹³ J — equivalent to burning 1.5 million kg of coal. This extreme energy density from mass-energy conversion (E = mc²) is what makes nuclear reactions so powerful compared to chemical reactions.
Related Free Tools on RoughTools
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Calculate Kinetic and Potential Energy Now
The free Energy Calculator at RoughTools calculates kinetic energy (½mv²), gravitational potential energy (mgh), spring elastic potential energy (½kx²), and work (F×d). Converts between Joules, kJ, kWh, calories, BTU, and foot-pounds. No account needed, completely free.