The work-energy theorem calculator computes work using W = Fd cos θ (work from force and displacement) or W = ΔKE = ½mv₂² − ½mv₁² (work from kinetic energy change). Enter any two known quantities to find the third.
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How to Use the Work-Energy Theorem Calculator
Work is one of the most fundamental concepts in physics — it measures the energy transferred when a force moves an object through a distance. This calculator handles two key formulas: work from force and displacement, and work from kinetic energy change.
Mode 1: Work from Force and Displacement (W = Fd cos θ)
Use this mode when you know the applied force and the displacement. The angle θ is the angle between the force vector and the direction of motion. For a person pushing a 50 lb box 10 feet across a floor with a force of 30 lb applied at 30° below horizontal:
W = 30 × 10 × cos(30°) = 300 × 0.866 = 259.8 ft·lb (≈ 352 J)
Practical cases: θ = 0° means force and motion are aligned (maximum work); θ = 90° means force is perpendicular to motion (zero work, like a satellite orbit); θ = 180° means force opposes motion (negative work, energy removed).
Mode 2: Work-Energy Theorem (W = ΔKE)
Use this mode when you know how an object's speed changed. The work-energy theorem states that net work equals the change in kinetic energy. For a 100 kg car accelerating from 0 to 27.8 m/s (about 100 km/h):
W = ½ × 100 × (27.8² − 0²) = ½ × 100 × 772.84 = 38,642 J ≈ 38.6 kJ
This is the minimum energy needed to accelerate the car — in reality, more energy is consumed overcoming friction and air resistance.
Solving for Force or Distance
Use the "Solve for" dropdown to find an unknown variable. If you know a motor does 500 J of work pulling a load 4 meters at 0°, the required force is F = W / (d × cos θ) = 500 / (4 × 1) = 125 N. The calculator rearranges the formula automatically.
Metric vs Imperial Units
Toggle between metric (N, m, m/s, joules) and imperial (lb, ft, ft/s, ft·lb). The calculator converts internally — 1 ft·lb = 1.35582 J. Physics problems from US textbooks often use imperial units; toggle to match your textbook.
For a detailed walkthrough, see our guide: Understanding Newton's Laws of Motion.
FAQ
What is the work-energy theorem?
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy: W_net = ΔKE = ½mv₂² - ½mv₁². If a 2 kg object speeds up from 3 m/s to 7 m/s, the net work done is ½(2)(49) - ½(2)(9) = 49 - 9 = 40 J.
What does the angle in W = Fd cos θ mean?
The angle θ is between the direction of the applied force and the direction of motion. If you push parallel to the ground (θ = 0°), all force contributes: cos 0° = 1. If you push perpendicular to motion (θ = 90°), no work is done: cos 90° = 0. Pushing at 60° means only half the force contributes to work: cos 60° = 0.5.
Can work be negative?
Yes. Negative work means energy is removed from the object. Friction does negative work on a sliding object — it acts opposite to motion (θ = 180°, cos 180° = -1). Gravity does negative work when lifting an object upward. The object loses kinetic energy when net work is negative.
What units is work measured in?
Work is measured in joules (J) in the SI system: 1 J = 1 N·m = 1 kg·m²/s². In imperial units, work is measured in foot-pounds (ft·lb): 1 ft·lb ≈ 1.356 J. Energy of 1 joule is roughly the work needed to lift a 100 g apple 1 meter.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted to any server.
What is the difference between work and power?
Work is the energy transferred by a force over a distance (W = Fd cos θ), measured in joules. Power is the rate at which work is done (P = W/t), measured in watts (1 W = 1 J/s). A motor that does 1000 J of work in 10 seconds has a power output of 100 W.