Work-energy theorem - Physics 1 AP Study Notes

The work-energy theorem is a fundamental principle stating that the net work done on an object equals its change in kinetic energy: Wnet = ΔKE = KEfinal − KEinitial.

Key Definitions:

• Work (W): Energy transferred when a force causes displacement. W = Fs cos θ, where F is force magnitude, s is displacement, and θ is the angle between force and displacement vectors. Unit: joule (J)
• Kinetic Energy (KE): Energy possessed by a moving object. KE = ½mv², where m is mass (kg) and v is velocity (m/s)
• Net Work: The algebraic sum of work done by all forces acting on an object

Essential Formula: Wnet = ½mv² − ½mu², where u is initial velocity and v is final velocity.

Critical Understanding: The theorem applies when calculating work from all forces combined. If multiple forces act (friction, applied force, gravity), sum their individual work contributions or use the net force directly.

Memory Aid (NEWTON): Net work Equals Total Work = Output kinetic energy minus Noriginal kinetic energy

Sign Conventions:

• Positive work increases kinetic energy (force and displacement in same direction)
• Negative work decreases kinetic energy (opposing motion, like friction)
• Zero work when force ⊥ displacement (θ = 90°)

Cambridge Syllabus Connection: This theorem bridges mechanics and energy concepts, appearing in questions involving variable forces, stopping distances, and energy transformations. Master this for Problem-solving and Analysis questions worth 6-8 marks.

How It Works: Imagine pushing a shopping trolley from rest. Your applied force does positive work, converting your chemical energy into the trolley's kinetic energy. When you stop pushing and friction acts alone, it does negative work, depleting the kinetic energy until the trolley stops.

Real-World Applications:

1. Vehicle Braking Systems: When a car traveling at 30 m/s brakes to rest, the work-energy theorem explains stopping distance. The brakes (friction force) must do negative work equal to the car's initial kinetic energy. Doubling speed quadruples kinetic energy (since KE ∝ v²), requiring four times the stopping distance—critical for road safety calculations.

2. Roller Coasters: As the coaster descends, gravity does positive work, increasing kinetic energy (speed). Climbing does negative work against gravity, decreasing speed. Energy conservation and work-energy theorem together explain why the first hill must be highest.

3. Archery: Drawing a bowstring does work on the arrow. The string's elastic potential energy converts to the arrow's kinetic energy upon release. Measuring draw force and distance allows calculating the arrow's launch velocity using W = ΔKE.

Powerful Analogy: Think of kinetic energy as money in your energy bank account. Work is deposits (positive) or withdrawals (negative). The work-energy theorem is simply your bank statement: final balance = initial balance + net transactions.

Conceptual Insight: Unlike momentum (conserved in isolated systems), kinetic energy changes whenever net work occurs. This makes the theorem invaluable for problems where forces vary with position or multiple forces act simultaneously.