The work-energy theorem connects forces, motion, and energy in one powerful idea: the net work done on an object equals its change in kinetic energy. It helps explain why a cart speeds up when pulled forward, slows down when friction dominates, or keeps the same speed when the net work is zero. This theorem is useful because it often avoids solving for acceleration and time directly.
Instead, it focuses on how forces acting over distances transfer energy to or from motion.
Work depends on the force, the displacement, and the angle between them, so only the part of a force along the motion changes kinetic energy. Positive net work increases speed, while negative net work decreases speed. If several forces act at once, their individual works add to give the net work.
In real situations such as carts, cars, roller coasters, and sliding blocks, the theorem provides a direct way to predict final speed from forces and distance.
Key Facts
- Work-energy theorem: W_net = ΔK
- Change in kinetic energy: ΔK = K_f - K_i
- Kinetic energy: K = 1/2 mv^2
- Work by a constant force: W = Fd cos θ
- Positive work occurs when a force has a component in the direction of displacement.
- Negative work occurs when a force has a component opposite the direction of displacement.
Vocabulary
- Work
- Work is the energy transferred by a force acting through a displacement.
- Net work
- Net work is the sum of the work done by all forces acting on an object.
- Kinetic energy
- Kinetic energy is the energy an object has because of its motion.
- Displacement
- Displacement is the straight-line change in position from an initial point to a final point.
- Friction
- Friction is a contact force that usually opposes motion and often does negative work.
Common Mistakes to Avoid
- Using total force instead of net force is wrong because the theorem uses the sum of all work done by all forces, not just the applied force.
- Forgetting the angle in W = Fd cos θ is wrong because only the component of force along the displacement does work.
- Treating kinetic energy as 1/2 mv instead of 1/2 mv^2 is wrong because speed is squared, so doubling speed makes kinetic energy four times larger.
- Assuming positive work always happens when a force is present is wrong because a force perpendicular to displacement does zero work and a force opposite displacement does negative work.
Practice Questions
- 1 A 5.0 kg cart starts from rest and is pulled with a constant net force of 20 N over 4.0 m. Use the work-energy theorem to find its final speed.
- 2 A 2.0 kg block moving at 6.0 m/s slides across a rough floor. Friction does 18 J of negative work on it. What is the block's final speed?
- 3 A student pushes a box across the floor at constant speed. Explain what the work-energy theorem says about the net work on the box and the change in its kinetic energy.