Work-Energy Theorem Calculator

Calculate net work done, change in kinetic energy, and velocity using the work-energy theorem.

0=force parallel to motion
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Net Work Changes Kinetic Energy

The work-energy theorem connects the net work done on an object to its change in translational kinetic energy. It is powerful because it avoids tracking acceleration at every instant: once all forces are combined through their work, Wnet=ΔK. Positive net work increases speed, negative net work decreases speed, and zero net work leaves kinetic energy unchanged even if individual forces are present.

Work is a scalar and depends on displacement along the force direction. A force perpendicular to motion can change direction without changing speed, so it may do no work. The theorem uses net work from all forces, including gravity, friction, applied forces, and normal forces when they have a component along the displacement.

Wnet=ΔK=½mvf2−½mvi2
SymbolMeaningWhy it appears / units
WnetNet workJ; algebraic sum of work by all forces.
mMasskg; appears in kinetic energy.
vi, vfInitial and final speedm/s; kinetic energy depends on speed squared.

Because kinetic energy depends on v², doubling speed requires four times as much kinetic energy. If net work is known, solve for the physically meaningful nonnegative speed; direction must be determined separately from the motion and forces.

Net positive work must increase kinetic energy. If Wnet>0, Kf>Ki; if net work is negative, kinetic energy decreases. For constant speed, the kinetic-energy change is zero, so the algebraic sum of all work must also be zero even when individual forces do nonzero work.

Worked Examples

Example 1: Push 2kg box: F=10N, d=5m, from rest
W=10×5=50J, KE₂=50J
Result: v=√(2×50/2)=7.07 m/s
Work-energy theorem directly
Example 2: F at 30°: 20N, d=3m, m=3kg, u=2m/s
W=20×3×cos30°=52J
Result: KE₂=6+52=58J, v=6.22m/s
Angled force component does work
Example 3: Braking a bicycle and rider
m=80kg, vi=10m/s, vf=0
Result: Wnet=−4000J
The negative sign means kinetic energy was removed from the moving system.
Example 4: Speed after positive work
m=2kg, vi=3m/s, Wnet=16J
Result: vf=5m/s
The energy method finds the final speed without first calculating acceleration or travel time.

Common Mistakes

⚠️
Using one force instead of net work

The theorem requires the sum of work from every force. Ignoring friction or gravity can give the wrong kinetic-energy change.

⚠️
Confusing speed with velocity

Kinetic energy contains v² and is not directional. The theorem determines speed changes; direction requires additional vector information.

⚠️
Forgetting the cosine in force work

For a constant force, W=Fd cosθ. A force perpendicular to displacement contributes zero work even if its magnitude is large.

Frequently Asked Questions

Work-energy theorem vs Newton's 2nd law?
Both give correct results. Work-energy (scalar, W=ΔKE) is easier when force and displacement are known. Newton's 2nd (vector) is better when finding acceleration or time.
Negative work?
When force component opposes displacement: friction, braking, gravity on uphill. Negative work removes kinetic energy. Net W=0 for circular motion (centripetal force always perpendicular to velocity).
Can the normal force do work?
Yes, depending on the situation. On a fixed horizontal surface it is perpendicular to horizontal displacement and does no work, but a moving surface or constrained geometry can allow a normal force to transfer energy.
Why is friction work usually negative?
Kinetic friction commonly acts opposite the relative sliding displacement, so the angle between force and displacement is 180° and Fd cos180° is negative. That reduces mechanical kinetic energy.
Does zero net work mean no forces act?
No. Forces can be present and cancel in their work contributions, or act perpendicular to motion. Uniform circular motion is a familiar case where centripetal force changes direction while doing no work on speed.
When is the work-energy theorem easier than F=ma?
It is especially useful when forces depend on position or when only initial and final speeds are needed. It replaces a time-based acceleration analysis with an energy balance over displacement.
Does the work–energy theorem use one force or the net force?
The theorem ΔK=Wnet uses the total work done by all forces on the object. Work by an individual force equals the full kinetic-energy change only when the other forces do zero net work or their work has been accounted for separately. Signs matter because opposing forces can do negative work.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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