Kinetic & Potential Energy Calculator
Kinetic & Potential Energy Calculator
Energy is the capacity to do work, and in classical mechanics it appears most often as kinetic energy (energy of motion) or potential energy (stored energy of position). Our Kinetic & Potential Energy Calculator computes three of the most common forms: translational kinetic energy, gravitational potential energy, and elastic potential energy stored in a stretched or compressed spring. These quantities are the building blocks of the work-energy theorem and conservation of energy, used everywhere from roller-coaster design to vehicle safety and mechanical engineering.
What Is Kinetic Energy?
Kinetic energy (KE) is the energy an object possesses because it is moving. A parked car has none; a moving car has a great deal, and doubling its speed quadruples its kinetic energy. The standard unit is the joule (J), where 1 J = 1 kg·m²/s².
What Is Potential Energy?
Potential energy is stored energy. Gravitational potential energy (GPE) depends on an object's height in a gravitational field; lift a book onto a shelf and you store energy that is released if it falls. Elastic potential energy (EPE) is stored when a spring or elastic material is deformed. Both are measured in joules.
A useful way to think about the distinction is "energy of motion" versus "energy of position or configuration." Kinetic energy belongs to an object that is already moving; potential energy belongs to an object that could move because of where it is or how it is shaped. A drawn bow, a raised weight, and a compressed gas cylinder all hold potential energy that converts to kinetic energy the instant the constraint is released. That conversion is the heart of nearly every mechanical system, from a pendulum swinging between KE and GPE to a watch spring unwinding from EPE into motion.
The calculator has three modes. Pick the energy type you need from the radio buttons, then enter the relevant quantities and click Calculate.
Kinetic Energy Mode
- Enter Mass in kilograms.
- Enter Velocity in meters per second.
- Click Calculate to get KE in joules.
Worked Example 1: A 70 kg Runner at 10 m/s
KE = ½ × 70 × 10² = 0.5 × 70 × 100 = 3500 J. The runner carries 3.5 kJ of kinetic energy.
Gravitational PE Mode
- Enter Mass in kg.
- Enter Height above a reference level in meters.
- Click Calculate. GPE = mgh with g = 9.80665 m/s².
Worked Example 2: A 70 kg Object Raised 5 m
GPE = 70 × 9.80665 × 5 ≈ 3432 J. Raising it 5 m stores about 3.43 kJ.
Elastic PE Mode
- Enter the Spring Constant (k) in N/m (stiffness).
- Enter the Displacement (x) from the spring's rest position in meters.
- Click Calculate. EPE = ½kx².
Worked Example 3: A Spring (k = 200 N/m) Stretched 0.5 m
EPE = ½ × 200 × 0.5² = 100 × 0.25 = 25 J. The stretched spring stores 25 J.
Edge Cases
- Zero mass or zero velocity → KE = 0 J (no motion, no energy).
- Zero height → GPE = 0 J at the reference level.
- Zero displacement → EPE = 0 J (unstretched spring).
- Negative inputs are rejected; energy magnitudes are non-negative here.
Worked Example 4: Comparing Energies of a Falling Ball
A 2 kg ball is held 10 m above the ground, then dropped (ignoring air resistance). At the top:
- GPE = 2 × 9.80665 × 10 ≈ 196.1 J, KE = 0 J.
At the instant before impact (height ≈ 0):
- GPE ≈ 0 J, KE = 196.1 J.
The total mechanical energy stays at 196.1 J throughout the fall — it simply changes form. This conservation principle is why the KE at impact equals the GPE lost, and it lets you predict impact speed: v = √(2·KE/m) = √(2 × 196.1 / 2) ≈ 14.0 m/s, matching the free-fall result.
Worked Example 5: A Car's KE at Highway Speed
A 1500 kg car at 30 m/s (about 108 km/h):
- KE = ½ × 1500 × 30² = 675,000 J = 675 kJ.
This is the energy that braking systems must dissipate to stop the car, and it scales so steeply with speed that a 60 m/s impact carries four times as much energy (2.7 MJ) for the same mass.
Translational kinetic energy of a mass m moving at speed v:
Gravitational potential energy near Earth's surface, with height h and gravity g:
Elastic potential energy in an ideal spring of constant k displaced by x:
These formulas assume non-relativistic speeds, a uniform gravitational field, and an ideal Hooke's-law spring.
They are also linked by the conservation of mechanical energy. In a closed system with no friction, KE + PE stays constant: any gain in one is exactly balanced by a loss in the other. For a mass sliding down a frictionless ramp, the GPE lost equals the KE gained, so the speed at the bottom depends only on the drop height, not the path taken. Similarly, when a spring launches a projectile, the EPE at full compression becomes KE at release. The work-energy theorem ties this together: the net work done on an object equals its change in kinetic energy, W_net = ΔKE.
Kinetic energy for a 70 kg mass at various speeds shows the quadratic dependence on velocity.
| Speed (m/s) | Kinetic Energy (J) |
|---|---|
| 2 | 140 |
| 5 | 875 |
| 10 | 3500 |
| 20 | 14000 |
| 30 | 31500 |
Notice that going from 10 to 20 m/s (doubling speed) multiplies KE by 4 (3500 → 14000 J).
Gravitational potential energy for a 70 kg mass at various heights:
| Height (m) | GPE (J) |
|---|---|
| 1 | 686 |
| 5 | 3432 |
| 10 | 6865 |
| 20 | 13729 |
| 50 | 34323 |
GPE grows linearly with height. Doubling the height doubles the stored energy.
Elastic potential energy for k = 200 N/m at various displacements:
| Displacement (m) | EPE (J) |
|---|---|
| 0.1 | 1.0 |
| 0.25 | 6.25 |
| 0.5 | 25.0 |
| 0.75 | 56.25 |
| 1.0 | 100.0 |
EPE grows with the square of displacement, so a spring stretched twice as far stores four times the energy.
A practical takeaway from all three tables: energy depends on squares (v² and x²) far more than on the linear terms (m and h). That is why a modest speed increase dominates a large mass increase when comparing moving vehicles, and why over-compressing a spring rapidly stores a great deal of energy that can snap back with force. When you size a safety system — a crumple zone, a bumper, a shock absorber — you are really sizing it against these squared quantities.
- Keep units consistent: kilograms, meters, seconds, and N/m. Mixing in grams or cm breaks the result.
- Remember the square: velocity and displacement appear squared, so small changes have large effects.
- Use g = 9.80665 for standard Earth calculations; use 1.62 on the Moon for comparison.
- Check reasonableness: a 70 kg person walking at 1.4 m/s has about 69 J — roughly the energy of a falling apple.
- For vehicles, KE helps estimate braking distance, since stopping work must equal the initial KE.
- Springs store more energy per unit displacement as k increases; a stiff spring needs more force for the same stretch.
- Pick a sensible reference height: GPE is only defined relative to a zero you choose. For a falling object, set the ground as zero; for a shelf, set the floor. Only the difference between two heights matters physically.
- Estimate impact from energy: if you know the drop height, GPE lost ≈ KE gained, so you can find impact speed without solving motion equations — handy for quick safety checks.
- Watch relativistic limits: at everyday speeds (well below 0.01% of light speed) KE = ½mv² is accurate to many decimal places; only particle-physics speeds need the relativistic correction.
- Combine modes carefully: a moving object at a height has both KE and GPE. Add them for total mechanical energy; don't double-count by also including EPE unless a spring is involved.
- Moon vs Earth: divide GPE by about 6 to compare the same lift on the lunar surface, since g is roughly 1.62 versus 9.81.
- Non-relativistic: KE = ½mv² fails near light speed, where relativistic formulas are required.
- Uniform gravity: GPE assumes constant g; over large altitudes g changes.
- Ideal springs: real springs deviate from Hooke's law when overstretched.
- No losses: friction, air drag, and hysteresis are ignored; real systems dissipate energy as heat.
- Reference level: GPE depends on the chosen zero-height; only differences are physically meaningful.
- Scalar, not vector: KE and the stored-energy magnitudes here are scalars. Direction of motion does not enter KE; only speed does.
- Thermal conversion: when a bouncing ball loses height each bounce, its "missing" mechanical energy became heat and sound, not vanished — energy is conserved overall, just not as useful mechanical energy.
- Speed upper bound: the formulas assume the object stays below a few percent of light speed; near-relativistic KE grows without the simple ½mv² bound.
What is kinetic energy measured in?
Joules (J). One joule equals one kilogram-meter squared per second squared (kg·m²/s²).
Why does doubling speed quadruple kinetic energy?
Because KE depends on v². If velocity doubles, v² becomes four times larger, so KE quadruples.
What value of g should I use?
The standard acceleration due to gravity, g = 9.80665 m/s², unless you are on another body (e.g., the Moon's g ≈ 1.62 m/s²).
Can potential energy be negative?
Gravitational PE can be negative if your reference height is above the object, but only differences matter physically. This calculator reports magnitudes using a zero at the reference level.
What is elastic potential energy?
It is the energy stored in a deformed elastic object, such as a stretched spring, given by ½kx².
How is spring constant measured?
In newtons per meter (N/m): the force needed to stretch the spring one meter. Stiffer springs have larger k.
Is kinetic energy conserved?
Total mechanical energy is conserved in isolated systems without friction, but KE alone can convert to PE and back (e.g., a swinging pendulum).
What is the work-energy theorem?
The net work done on an object equals its change in kinetic energy: W = ΔKE.
Why does a heavier object have more KE at the same speed?
KE is proportional to mass m, so doubling the mass doubles the kinetic energy for the same velocity.
Can I calculate energy for rotating objects?
Rotational kinetic energy uses ½Iω² (moment of inertia times angular speed squared), which is a separate formula not covered by this translational calculator.
What is the difference between energy and power?
Energy is the total capacity to do work (measured in joules); power is the rate at which energy is delivered or used (watts = joules per second). A 100 J spring released in 1 s delivers 100 W; released in 0.1 s it briefly delivers 1000 W.
How much energy is in a falling object?
The KE at impact equals the GPE it lost: mgh. So a 1 kg object dropped 1 m carries about 9.8 J at impact, regardless of the path it took to get there.
Can potential energy be stored without height?
Yes — elastic potential energy (½kx²) is stored by deforming a spring, and chemical potential energy is stored in molecular bonds, neither of which requires height.
Why do vehicle safety features focus on KE?
Because KE = ½mv² grows with the square of speed, so the energy a crash system must absorb rises dramatically with speed. Slowing from 30 to 15 m/s cuts the KE to one quarter.
Last updated: July 18, 2026
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