Question

Explain the work-energy theorem, including the direction energy moves and how it relates to positive and negative work.

Answer 1

Summary statement
- The work–energy theorem: the net work done on an object by all forces equals the change in its kinetic energy.
W_net = ΔK = K_final − K_initial = 1/2 m v_f^2 − 1/2 m v_i^2.

Short derivation (sketch)
- From Newton’s second law F = ma, dot both sides with velocity v:
F · v = m a · v = d/dt (1/2 m v^2).
- Integrate over time (or integrate F · ds over the path) to get
∫ F · ds = Δ(1/2 m v^2) → W_net = ΔK.

What “work” means and how direction matters
- Work by a force is W = ∫ F · ds. For a constant force this is W = F d cosθ, where θ is the angle between force and displacement.
- If θ = 0° (force along displacement) cosθ = 1 → positive work.
- If θ = 180° (force opposite displacement) cosθ = −1 → negative work.
- If θ = 90° (force perpendicular to displacement) cosθ = 0 → zero work.
- Positive work transfers energy into the object’s kinetic energy (K increases).
- Negative work extracts kinetic energy from the object (K decreases); the removed energy goes into whatever receives it (heat, internal energy, potential energy in a spring, work done on another object, etc.).

Relation between positive/negative work, energy flow, and types of forces
- Net positive work: energy flows into the object’s kinetic store.
- Net negative work: energy flows out of the object’s kinetic store.
- Individual forces can do positive or negative work; only the sum (net work) determines ΔK.
- For conservative forces (e.g., gravity, spring) work can be associated with a change in potential energy: W_cons = −ΔU. Combining with the theorem gives energy conservation when nonconservative work is zero: ΔK + ΔU = 0.
- Nonconservative forces (friction) do negative work and convert mechanical energy to internal/thermal energy. In general: W_nonconservative = ΔK + ΔU (or equivalently ΔE_mech = W_nonconservative).

Examples
- Pushing a stationary block so it speeds up: the push does positive work → KE increases.
- Braking a car (friction at brakes/tires): friction does negative work on the car → KE decreases and becomes heat.
- Object moving in a horizontal circle at constant speed: centripetal force is perpendicular to motion → does zero work, KE unchanged.

Units
- Work and kinetic energy have units of joules (J).

Bottom line
- Work is energy transferred by forces through displacement. The sign of the work tells whether kinetic energy is gained (positive work) or lost (negative work); the net work on an object exactly equals its change in kinetic energy.