The energy of a body is defined as its capacity for doing work. Energy is a scalar quantity with dimension [ML²T⁻²] and SI unit Joule. It can be transformed from one form to another but cannot be created or destroyed.

• Energy and work have the same dimensions and units — energy is simply stored capacity to do work.
• Mechanical energy = KE + PE; total mechanical energy is conserved in systems with only conservative forces.
• Common NEET trap: confusing energy (capacity to do work) with power (rate of doing work) — energy is in Joules, power in Watts.

The energy possessed by a body by virtue of its motion is called kinetic energy. KE = ½mv² = p²/2m, where p = mv is the linear momentum. This work done appears as the kinetic energy of the body.

• KE = p²/2m: NEET gives momentum p instead of velocity v — use this form directly to avoid computing v first.
• KE is always non-negative (v² ≥ 0 always); it can never be negative unlike potential energy.
• If the speed of a body is doubled, KE increases by a factor of 4 (since KE ∝ v²); if momentum is doubled, KE increases by a factor of 4 as well (since p² is in the numerator).

When a vehicle moving at speed v decelerates under a constant retarding force F, the stopping distance s = mv²/2F = KE/F. Derived by: F × s = ΔKE = ½mv² (initial KE is fully converted to heat by braking force).

• Stopping distance ∝ v² — doubling speed quadruples the stopping distance at the same retarding force.
• Stopping distance ∝ 1/F — halving the brake force (wet road) doubles the stopping distance.
• Common NEET scenario: two vehicles same mass but different speeds — the faster one needs proportionally more stopping distance, which NEET asks as a ratio problem.

Potential energy is the energy possessed by a body by virtue of its position or configuration. W = −ΔPE: work done by a conservative force equals the decrease in potential energy. U = mgh (gravitational), U = ½kx² (elastic).

• Potential energy is always relative to a chosen reference position where PE = 0.
• A body gains PE when work is done against a conservative force; it releases PE when the conservative force does positive work.
• Common NEET trap: using the wrong reference level for gravitational PE — always state the reference clearly before computing PE differences.

Equilibrium: net force = 0 (dPE/dx = 0). Stable: PE minimum (d²PE/dx² > 0) — body returns after displacement. Unstable: PE maximum (d²PE/dx² < 0) — body moves further after displacement. Neutral: PE constant (d²PE/dx² = 0) — body stays in new position.

• NEET often shows a PE vs x curve with labelled points and asks to identify equilibrium type — look for minimum (stable), maximum (unstable), and flat region (neutral).
• Physical examples: ball in a bowl (stable), ball on top of a hill (unstable), ball on a flat table (neutral).
• Common NEET trap: confusing equilibrium of forces (ΣF=0) with stable equilibrium — all equilibrium positions have ΣF=0, but only stable equilibrium lies at PE minimum.

When a spring of spring constant k is compressed or stretched by x, elastic PE = ½kx². Work done by the spring force when displaced from x₁ to x₂: W = ½kx₁² − ½kx₂².

• The spring force is conservative — elastic PE is fully recoverable when the spring returns to its natural length.
• Elastic PE = ½kx² for both extension and compression; PE is always positive regardless of the sign of x.
• Common NEET trap: computing W = kx × x = kx² instead of ½kx² for work done by a spring — the ½ factor comes from the force increasing linearly from 0 to kx.

Electrical PE of a charge Q in a potential V: PE = QV. For a system of two point charges: PE = kQ₁Q₂/r. Work done to assemble a charge configuration equals the electrical PE stored.

• Electrical PE is positive for like charges (repulsion — positive work needed to bring them together) and negative for unlike charges (attraction).
• The reference for electrical PE is usually r → ∞ where PE = 0.
• NEET tests electrical PE mostly in the context of charged particle motion in a uniform electric field, where W = QEd gives the gain in KE.

Near the Earth's surface, gravitational PE = mgh where h is the height above the reference level. For large distances: PE = −GMm/r (negative because gravity is attractive). The work done against gravity to lift a mass = mgh.

• Gravitational PE is always negative in the universal formula (−GMm/r) because the reference is PE=0 at r→∞ and gravity is an attractive force.
• The gain in gravitational PE when lifting a mass m by height h is mgh, regardless of the path taken (gravity is conservative).
• Common NEET trap: using U = mgh for large-distance orbital calculations — this formula is an approximation valid only near Earth's surface where g ≈ constant.

For a uniform chain of mass M and length L with one end hanging off a table: work done to pull the hanging part (length l) onto the table = Mgl²/2L (the CM of the hanging part rises by l/2).

• Treat the hanging portion as a mass (Ml/L) whose centre of mass is at a depth l/2 below the table edge.
• Work done = (mass of hanging part) × g × (rise of CM) = (Ml/L) × g × (l/2) = Mgl²/2L.
• Common NEET trap: using the full chain length L instead of the hanging length l in the formula — the formula uses l² where l is the portion that was hanging.

For a uniform chain of mass M and length L falling off a table, the velocity when the entire chain just leaves the table: v = sqrt(gL) (using energy conservation; the CM falls by L/2 from initial position).

• Initial state: chain just begins to fall (v=0); the CM of the hanging part is at a depth of L/2 below the table (for the full chain initially hanging).
• Energy conservation: mgh = ½mv² → gL = v² (wait — h = L/2 for CM elevation drop, but this depends on the initial condition); use W_gravity = ½Mv² for the full problem.
• The general result for velocity when length x has left the table: v = sqrt(gx) — NEET uses this in numerical form.

Energy can neither be created nor destroyed; it can only be converted from one form to another. In a conservative system: total mechanical energy E = KE + PE = constant at all points. In the presence of friction: E_initial = E_final + W_friction.

• The total energy of the universe is constant — only its form changes (chemical → thermal, kinetic → potential, etc.).
• Conservation of mechanical energy holds only when no non-conservative forces (friction, air drag) do work on the system.
• Common NEET trap: applying mechanical energy conservation when friction is present — include W_friction as an energy loss: ½mv₂² + mgh₂ = ½mv₁² + mgh₁ − W_friction.