Work Energy Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work — When Force Becomes Energy:

Work done by a force is defined as W = ∫F·ds = Fs cosθ, where θ is the angle between the force direction and the direction of displacement. Work is a scalar quantity (though force and displacement are vectors). SI unit: Joule (J = N·m). Work can be positive (force aids motion, 0° ≤ θ < 90°), zero (force perpendicular to motion, θ = 90°), or negative (force opposes motion, 90° < θ ≤ 180°).

Kinetic Energy: KE = ½mv². This is always non-negative and increases with the square of velocity — doubling speed quadruples kinetic energy. KE is frame-dependent: a passenger in a car has KE = 0 relative to the car but KE = ½mv_car² relative to the ground.

Potential Energy: PE = mgh (gravitational, near Earth’s surface where g is constant); PE = ½kx² (spring, where x is displacement from natural length). Potential energy is defined up to an arbitrary constant — what matters is the change in PE, not its absolute value.

Power: P = W/t (average power) = F·v (instantaneous power when force F acts on a body moving with velocity v). SI unit: Watt (W = J/s). 1 horsepower ≈ 746 W. For vehicles, power determines maximum speed on level ground (P = F_total × v_max when acceleration = 0, so v_max = P/friction).

⚡ JEE Tip: Work-Energy theorem W_net = ΔKE is valid in ALL situations — even with friction, even with variable forces, even in non-inertial frames (if pseudo forces are included in W_net). It is often easier than SUVAT equations when work done by individual forces can be computed.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Conservative vs Non-Conservative Forces:

A force is conservative if the work done moving a body between two points is independent of the path taken. Equivalent conditions: (1) Work done in a closed loop = 0, (2) A potential energy function U(x) exists such that F = -dU/dx. Gravitational force (F = -mg j), spring force (F = -kx i), and electrostatic force are conservative.

Non-conservative forces (friction, air resistance, tension in a rope with dissipation) change mechanical energy into heat, sound, or other forms. For these forces, W_net = ΔKE still holds, but mechanical energy is NOT conserved.

Collisions — Classification:

Type	Coefficient of Restitution (e)	KE Conservation	Example
Perfectly elastic	e = 1	KE conserved	Billiard balls, atomic collisions
Partially elastic	0 < e < 1	Some KE lost	Real-world drops, football
Perfectly inelastic	e = 0	Maximum KE lost (objects stick)	Ball of clay hitting floor

e = v_separation/v_approach = √(KE_lost_during_collision/KE_before_collision) in 1D.

One-Dimensional Elastic Collision Formulas: For a mass m₁ moving at velocity u₁ striking stationary mass m₂ (u₂ = 0):

• Final velocity of m₁: v₁ = ((m₁ - m₂)/(m₁ + m₂)) × u₁
• Final velocity of m₂: v₂ = (2m₁/(m₁ + m₂)) × u₁

If m₁ = m₂: v₁ = 0, v₂ = u₁ — the moving mass stops and the target mass takes all the velocity (Newton’s cradle).

⚡ JEE Tip: For two-body problems in JEE, use the Centre of Mass frame. In the CoM frame, total momentum = 0. Kinetic energy in the lab frame = KE_CoM + KE_rel (relative kinetic energy in CoM frame). For elastic collisions, KE_rel is conserved; for perfectly inelastic collisions, KE_rel = 0 after collision.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Variable Force Work — Integration Approach:

When force varies with position, F = F(x), work is calculated by integrating: W = ∫(from x₁ to x₂) F(x) dx. Graphically, work done = area under the F-x curve. For a spring, W_spring = ∫(0 to x) kx’ dx’ = ½kx². For gravitational force varying with distance (F = GMm/r², direction toward centre), work from r₁ to r₂: W = GMm(1/r₂ - 1/r₁).

Vertical Circular Motion — Complete Analysis:

For a mass tied to a string undergoing vertical circular motion:

• At the top of the circle: minimum speed v_top = √(gR) for the string to remain taut (centripetal force ≥ gravitational force: mv²/R ≥ mg)
• Minimum speed at the bottom to complete the circle: v_bottom ≥ √(5gR). This comes from energy conservation: KE_bottom - KE_top = 2mgR (height difference = 2R), so ½mv_bottom² - ½mv_top² = 2mgR. With v_top = √(gR), v_bottom = √(5gR + 2gR) = √(7gR)… wait, let’s recalculate: KE_top = ½m(gR), KE_bottom = ½mv². Height difference h = 2R. Energy: ½mv² = ½mgR + 2mgR = 2.5mgR = 5/2 mgR, so v² = 5gR, v = √(5gR). Yes.

Rocket Propulsion — Tsiolkovsky Equation:

A rocket ejects fuel at relative velocity v_e (exhaust velocity) with respect to the rocket. During a small time dt, fuel of mass dm is expelled, rocket mass changes by dm (negative). Momentum conservation in the rocket frame: mdv = v_e dm. Rearranging and integrating: Δv = v_e ln(m₀/m_f), where m₀ is initial mass and m_f is final mass (when fuel is exhausted). This is the Tsiolkovsky rocket equation. Key insight: the mass ratio m₀/m_f determines the achievable Δv, independent of burn time.

⚡ JEE Advanced Pattern: The centre of mass (CoM) frame is crucial for 2D collision problems. In the lab frame, two equal masses colliding elastically at 90° scatter at 90° to each other (because momentum components must be equal and kinetic energy is split equally). JEE Advanced 2022 Qn 41 involved a neutron-proton elastic collision at an angle — students needed to apply conservation of momentum in 2D (x and y separately) and conservation of kinetic energy to find the scattering angles. Work-energy-power problems in JEE Advanced frequently integrate with rotational motion (rolling spheres), thermodynamics (Carnot cycle analysis), and electrostatics (potential energy of charge configurations).