Work, Energy and Power

The Work-Energy theorem states:

W_net = ΔKE = ½m(v² − u²)

In words: the net work done on an object equals its change in kinetic energy.

Why it's powerful. It bypasses time entirely. If a problem gives you forces and distances but not time, work-energy is faster than F = ma + kinematic equations.

When to use it:
• You have force(s) acting over a known distance and want the final speed.
• Friction acts over a path of known length.
• A spring compresses/extends and you want the resulting velocity.

When not to use it:
• Time is a key unknown (use F = ma + v = u + at instead).
• Direction matters (KE is scalar — you lose direction info).
• Multiple objects with constraints (use momentum / Lagrangian).

Worked example. A 2 kg ball at rest is pushed horizontally by 10 N for 5 m on a frictionless surface. Net work = 10 × 5 = 50 J. So ½ × 2 × v² = 50 → v² = 50 → v = √50 ≈ 7.07 m/s.

Mechanical energy = KE + PE. When only conservative forces (gravity, springs) act on a system, the total mechanical energy is conserved:

KE_i + PE_i = KE_f + PE_f

Conservative forces: work done depends only on starting and ending positions, not the path. Gravity and springs are conservative.

Non-conservative forces: friction, drag, applied force. Work done depends on the path. When these act, mechanical energy is not conserved — but you can still apply:

ΔKE + ΔPE = W_nc (work done by non-conservative forces).

Worked example. A 1 kg ball is dropped from height 5 m. Just before hitting the ground, all PE has converted to KE (ignoring air resistance). PE_i = mgh = 1 × 9.8 × 5 = 49 J. So KE_f = 49 J → ½ × 1 × v² = 49 → v ≈ 9.9 m/s.

If air resistance does −10 J of work, then KE_f = 49 − 10 = 39 J → v ≈ 8.83 m/s. The "missing" 10 J becomes heat and sound.