What is the Work–Energy Theorem?
The work–energy theorem connects force and motion: the net work done on an object equals its change in kinetic energy. It's one of the most useful shortcuts in mechanics, letting you skip acceleration entirely.
The work–energy theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv_f² − ½mv_i².
- 1↓Net force appliedAn unbalanced force acts on the object, causing acceleration.
- 2↓Displacement occursThe object moves a distance d in the direction of the net force.
- 3↓Work is doneWork equals force times displacement: W = F·d.
- 4Kinetic energy changesThat work exactly equals the object's change in kinetic energy: W = ΔKE.
Try it: interactive calculator
Step-by-step worked examples
A 10 kg object starts at rest and reaches 6 m/s. Find the net work done on it.
W = ½mv_f² − ½mv_i² W = ½×10×6² − 0 = ½×10×36 W = 180 J
A 2 kg ball moving at 8 m/s slows to 3 m/s. Find the work done by friction.
W = ½mv_f² − ½mv_i² W = ½×2×3² − ½×2×8² = 9 − 64 W = −55 J (friction removes 55 J)
A constant net force of 20 N pushes a 4 kg crate 5 m, starting from rest. Use W = F·d to verify with the work–energy theorem.
W = F×d = 20×5 = 100 J ΔKE = ½mv_f² → 100 = ½×4×v_f² → v_f² = 50 → v_f ≈ 7.07 m/s
Flashcards
Quick quiz
Q1.A 5 kg object speeds up from 0 to 4 m/s. What is the net work done?
Q2.The work–energy theorem relates net work to...
Q3.If net work done on an object is negative, the object...
Q4.A 2 kg object moving at 10 m/s comes to rest. How much work was done on it?
The full card deck, worked steps and AI-tutor support for “What is the Work–Energy Theorem?” are in Notek — study by hand before your exam.
Common mistakes
Forgetting to square the velocities. — Correct: Kinetic energy uses v², not v — always square before subtracting.
Assuming work is always positive. — Correct: Work can be negative when force opposes displacement (e.g. friction, braking).
Using only the applied force instead of net force. — Correct: The theorem requires the NET force — sum of all forces acting on the object.
Confusing the work–energy theorem with conservation of energy. — Correct: The work–energy theorem relates net work to kinetic energy change only; it doesn't require energy conservation of the whole system.
FAQ
What is the formula for the work–energy theorem?
W_net = ΔKE = ½mv_f² − ½mv_i², where W_net is the total work done by all forces.
What is the work–energy theorem used for?
It's used to find velocity, work, or kinetic energy changes without needing acceleration or time.
What are examples of the work–energy theorem?
A car braking to a stop, a ball rolling up a hill, and a bat hitting a baseball all follow the work–energy theorem.
How do you calculate work using the work–energy theorem?
Calculate the object's kinetic energy at the start and end, then subtract: W = KE_final − KE_initial.