Work Energy Theorem Calculator
Calculate the net work done on an object as its change in kinetic energy, from its mass and its initial and final velocities.
Work Energy Theorem Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: W = ΔKE = ½mv₂² - ½mv₁²
Worked example — 1000 J
Formula
W = ΔKE = ½mv₂² - ½mv₁²
The net work done on an object equals its change in kinetic energy.
Worked Examples
Example 1: Accelerating object
Problem:10 kg, v₁=5 m/s, v₂=15 m/s
Solution:W=½(10)(225-25)=1000 J
Result:1000 J
Frequently Asked Questions
What is the work-energy theorem?
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv₂² − ½mv₁². This means any net force that accelerates an object transfers energy to it, while a decelerating force removes energy.
How do you calculate net work using the work-energy theorem?
Compute KE₁ = ½mv₁² and KE₂ = ½mv₂², then W = KE₂ − KE₁. Example: a 10 kg object accelerates from 5 m/s to 15 m/s. KE₁ = 0.5×10×25 = 125 J, KE₂ = 0.5×10×225 = 1125 J, so W = 1125 − 125 = 1000 J.
What units are used in work-energy calculations?
Mass m is in kilograms (kg), velocities v₁ and v₂ in metres per second (m/s). Kinetic energy KE = ½mv² is in Joules (J). Work W is also in Joules (J). One Joule equals one Newton-metre (1 J = 1 N·m = 1 kg·m²/s²).
What does negative work mean in the work-energy theorem?
Negative work (W < 0) means the net force opposes the motion, reducing kinetic energy. For example, friction does negative work: a 5 kg box sliding on a rough surface from 10 m/s to rest loses KE = ½×5×100 = 250 J, so friction did −250 J of work on the box.
What is the difference between the work-energy theorem and conservation of mechanical energy?
The work-energy theorem always holds: net work from every force, friction and drag included, equals ΔKE. Conservation of mechanical energy (KE + PE constant) is the narrower special case that applies only when non-conservative forces do no work. Use W = ΔKE when a rough surface, brake or air resistance is present, and switch to KE + PE bookkeeping for smooth ramps, pendulums and free fall.
Why does doubling an object's speed quadruple the work needed to stop it?
Kinetic energy carries v squared, so KE = ½mv² grows with the square of speed. Doubling v from 10 m/s to 20 m/s multiplies KE by four, and the theorem says four times the net work must be removed. With a constant braking friction force the stopping distance follows d = v² / (2μg): at μ = 0.7 and g = 9.81 m/s², 20 m/s needs 400 / 13.73 ≈ 29 m while 10 m/s needs only about 7.3 m.
How do you find the average net force from a change in kinetic energy?
Rearrange W = F_avg × d to get F_avg = ΔKE / d, valid when the force acts along the displacement. If 1000 J of net work is delivered over 4 m, the average net force is 1000 / 4 = 250 N. This is the standard way to estimate crash or impact forces: measure the speed lost and the crush distance, and the theorem returns the mean force without any force sensor.
Does the theorem still apply when the force varies during the motion?
Yes. Net work is defined as the integral of force along the path, W = ∫F·dx, so a spring force that grows with extension or a drag force that falls as the object slows is fully accounted for. The theorem cares only about the two kinetic-energy endpoints, which is exactly why it shortcuts problems where the intermediate force history is unknown or too messy to integrate.
Why can a force act on an object and still do zero net work?
Work depends on the component of force along the displacement, W = F·d·cos(θ), so a force perpendicular to the velocity contributes nothing. In uniform circular motion the centripetal force continuously changes direction but not speed, leaving v² and therefore KE unchanged. Because kinetic energy is a scalar built from speed alone, Work Energy Theorem Calculator needs no direction information for v₁ and v₂.
How does gravity enter a work-energy calculation for a falling object?
Over a drop of height h, gravity does W = mgh of positive work, so ½mv₂² − ½mv₁² = mgh. Released from rest this gives v = √(2gh): from 5 m, v = √(2 × 9.81 × 5) = √98.1 ≈ 9.90 m/s, independent of mass. If air resistance is significant its negative work must be subtracted, which is why heavy compact objects reach the ground faster than light bulky ones.
Background & Theory
Why Skip Straight to Kinetic Energy?
The work-energy theorem is a shortcut: instead of tracking the exact force profile an object experiences over a distance (which can be messy for varying forces like air resistance or a crumpling car body), you only need the mass and the speed before and after. The net work done by ALL forces combined equals the change in kinetic energy — full stop, regardless of how complicated the underlying force history was.
| Scenario | Sign of net work | What it means |
|---|---|---|
| Car accelerating from a stoplight | Positive | Engine adds kinetic energy |
| Car braking to a stop | Negative | Brakes/friction remove kinetic energy, converting it to heat |
| Crumple zone absorbing a crash | Large negative | Deformation work dissipates kinetic energy over a longer stopping distance, lowering peak force |
| Satellite in stable circular orbit | Zero | Gravity acts perpendicular to velocity, so it does no work and speed stays constant |
This is exactly why crumple zones work: spreading a crash's negative work over a longer collapse distance reduces the average braking force for the same total change in kinetic energy (W = F x d, so a bigger d for the same W means a smaller F).
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Specific Work Compressor Calculator
Calculate specific work compressor with inputs, formulas, and instant results.
🧮Work Calculator
Calculate work with inputs, formulas, and instant results.
🧮Rotational Kinetic Energy Calculator
Calculate rotational kinetic energy with inputs, formulas, and instant results.
🧮Capacitor Energy Calculator
Calculate capacitor energy with inputs, formulas, and instant results.
🧮Potential Energy Calculator
Calculate potential energy with inputs, formulas, and instant results.
🧮Photon Energy Calculator
photon energy calculator. Get instant, accurate results.
🧮Relativistic Energy Calculator
relativistic energy calculator. Get instant, accurate results.
⚡Kinetic Energy
Calculate energy of motion