Work, Energy & Power — vidhyapath 2027

Concept Core

Work, kinetic energy, potential energy, conservation — and when to use each.

Work Done by a Force

Work = Force × displacement component in direction of force.

W = F · d = Fd cosθ

θ is angle between F and displacement. Work is a scalar.

W > 0: force aids motion. W < 0: force opposes motion (friction, braking). W = 0: force ⊥ displacement (normal force, centripetal force).

Kinetic Energy & the Work-Energy Theorem

KE = ½mv². The Work-Energy Theorem is one of the most powerful tools in mechanics:

W_net = ΔKE = ½mv² − ½mu²

Net work done on an object = change in its kinetic energy. Use when forces are complex — no need for acceleration.

Potential Energy

Gravitational PE: U = mgh (h = height above reference)

Elastic PE (spring): U = ½kx² (k = spring constant, x = compression/extension)

PE is the energy stored by virtue of position or configuration. It depends on the reference level — only changes in PE are physical.

Conservation of Mechanical Energy

When only conservative forces do work (gravity, spring — NOT friction):

KE + PE = constant ½mv₁² + mgh₁ = ½mv₂² + mgh₂

This is the "no-friction" shortcut. Height and velocity trade off smoothly.

Power

Rate of doing work:

P = W/t = F·v = Fv cosθ

Unit: Watt (W) = J/s. 1 HP = 746 W.

When a machine operates at constant power: P = Fv. As speed increases, the force the engine can exert decreases (for fixed P).

Collision Types

Elastic: Both momentum AND KE conserved. e = 1 (coefficient of restitution).

Inelastic: Only momentum conserved. KE lost as heat. e < 1.

Perfectly inelastic: Bodies stick together. Maximum KE loss. e = 0.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (always)

Spring Problems — Elastic PE

Spring compressed by x stores energy U = ½kx². When released, all PE converts to KE:

½kx² = ½mv² → v = x√(k/m)

For two springs in series: 1/k_eff = 1/k₁ + 1/k₂ (softer combined). In parallel: k_eff = k₁ + k₂ (stiffer combined).

Formula Vault

All Work-Energy-Power formulas — exam-ready.

Work Done

W = Fd cosθ

θ = angle between F and d

Kinetic Energy

KE = ½mv²

Always ≥ 0

Work-Energy Theorem

W_net = ΔKE = ½mv²−½mu²

Net work = change in KE

Gravitational PE

U = mgh

h from reference level

Spring PE

U = ½kx²

k = spring constant; x = deformation

Conservation of Energy

KE₁ + PE₁ = KE₂ + PE₂

No friction; conservative forces only

Power

P = W/t = Fv cosθ

1 HP = 746 W

Elastic Collision Velocities

v₁ = (m₁−m₂)u₁+2m₂u₂)/(m₁+m₂)

Similarly for v₂

Perfectly Inelastic

v = (m₁u₁+m₂u₂)/(m₁+m₂)

Bodies stick together

KE lost in P.I. Collision

ΔKE = m₁m₂(u₁−u₂)²/2(m₁+m₂)

Always positive (lost)

Worked Examples

5 problems showcasing when Work-Energy beats Newton's Laws.

EasyFind velocity of 2 kg block after 10 J of work done, starting from rest▾

A 2 kg block starts from rest. Net work done on it = 10 J. Find its final velocity.

W_net = ΔKE → 10 = ½(2)v² − 0 → v² = 10 → v = √10 ≈ 3.16 m/s

✓ v = √10 m/s

MediumBall dropped from 10 m — find velocity just before hitting ground▾

A 1 kg ball is dropped from height 10 m. Using conservation of energy, find velocity just before impact. (g = 10)

At top: KE = 0, PE = mgh = 1×10×10 = 100 J. At bottom: PE = 0.

Conservation: ½mv² = 100 → v² = 200 → v = 10√2 ≈ 14.1 m/s

✓ v = 10√2 m/s (same as v = √(2gh) = √200)

MediumSpring compressed 0.2 m launches 0.5 kg block — find speed▾

A spring (k = 800 N/m) is compressed by 0.2 m and releases a 0.5 kg block on a frictionless surface. Find the block's speed after release.

Spring PE → KE: ½kx² = ½mv² → kx² = mv²

v² = kx²/m = 800×0.04/0.5 = 32/0.5 = 64 → v = 8 m/s

✓ Speed = 8 m/s

EAPCET LevelPerfectly inelastic collision — find KE lost▾

A 4 kg block moving at 6 m/s collides and sticks to a 2 kg block at rest. Find the KE lost.

Momentum conservation: 4×6 + 2×0 = (4+2)v → v = 24/6 = 4 m/s

Initial KE = ½×4×36 = 72 J

Final KE = ½×6×16 = 48 J

KE lost = 72 − 48 = 24 J

✓ KE lost = 24 J

Trap QuestionEngine at constant power — does velocity keep increasing forever?▾

An engine of power P drives a car against drag force f. What is the maximum speed? ⚠️ Conceptual trap.

The trap: Students think constant power → constant acceleration → speed increases forever. Wrong.

P = Fv. As v increases, the available engine force F = P/v decreases.

Maximum speed when engine force = drag force: P/v_max = f → v_max = P/f

At v_max, net force = 0 → acceleration = 0 → constant speed. The car reaches terminal velocity.

✓ Maximum speed = P/f where f = drag force

Mistake DNA

4 errors from distractor analysis — conceptual and computational.

📐

Work Done by Normal Force and Gravity While Moving Horizontally

Students sometimes add mgh to a horizontal motion problem or claim N does work on a horizontal surface.

❌ Wrong

Block moves 5 m horizontally:
W_gravity = mgh ≠ 0 ✗
W_normal = Fd ≠ 0 ✗

✓ Correct

Horizontal motion, h=0:
W_gravity = mg×0 = 0 ✓
W_normal = N×d×cos90° = 0 ✓

Work = Fd cosθ. Normal force ⊥ motion → θ=90° → W=0. Gravity does work only when there's vertical displacement.

🔋

Using Conservation of Energy When Friction is Present

Conservation of mechanical energy requires no non-conservative forces. Friction violates it.

❌ Wrong

With friction μ on incline:
mgh = ½mv² ✗
(ignores friction work)

✓ Correct

mgh − W_friction = ½mv²
W_friction = μmgcosθ × d ✓
Energy lost to friction

With friction: use W_net = ΔKE (Work-Energy Theorem). W_net = W_gravity + W_friction = ΔKE. Never ignore friction in energy problems.

💥

Assuming KE is Conserved in All Collisions

KE is conserved only in elastic collisions. Momentum is always conserved.

❌ Wrong

Inelastic collision:
Using ½m₁u₁² = ½(m₁+m₂)v²
to find v ✗

✓ Correct

For inelastic: only
momentum conserved:
m₁u₁ = (m₁+m₂)v ✓

Always identify collision type first. "Sticks together" = perfectly inelastic. Use momentum equation to find v, then compute KE if asked.

⚡

Power = Force × Velocity Confusion with Direction

When force and velocity are at an angle, P = Fv cosθ. Students forget the cosθ.

❌ Wrong

Force 100 N at 60° to
motion, v = 5 m/s:
P = 100 × 5 = 500 W ✗

✓ Correct

P = Fv cosθ
= 100 × 5 × cos60°
= 500 × 0.5 = 250 W ✓

Power is the dot product F⃗·v⃗ = Fv cosθ. Only the component of force along velocity contributes to power.

Chapter Intelligence

Work-Energy is a bridge chapter — it connects kinematics to everything ahead.

EAPCET Topic Weightage (2019–2024)

Work-Energy theorem

Conservation of energy

Collisions

Spring problems

Power calculations

High-Yield PYQ Patterns

Speed using energy conservation KE lost in inelastic collision Spring releases block — find speed Max speed at constant power Work done against friction Ball on incline — energy method

Exam Strategy

When a question involves height and velocity — try energy conservation first. It's faster than Newton's 2nd Law + kinematics.
For collision questions: identify elastic or inelastic first. Perfectly inelastic (stick together) → momentum conservation only.
Maximum speed at constant power: set F_engine = F_drag → v_max = P/F_drag. This appears almost every year.
Spring questions: energy stored = ½kx². If the block slides on a frictionless surface and the spring releases, all spring PE → KE.
Work done by friction is always negative. W_friction = −μmgcosθ × d. This energy is "lost" (converted to heat).