Board exam focus — Work, Energy and Power (CBSE & HBSE)

This chapter develops the concepts of work done by constant and variable forces, kinetic energy and the work-energy theorem, potential energy and the conservation of mechanical energy, power, and collisions in one and two dimensions. It is a conceptually rich and numerically intensive chapter that is highly scoring in both CBSE and HBSE Class 11 examinations, linking force-based dynamics to energy-based analysis.

Work and the Work-Energy Theorem

Work Done by a Constant Force

Work is said to be done when a force produces displacement in its own direction. For a constant force F producing displacement d at angle θ between them:

$$W = \vec{F}\cdot\vec{d} = Fd\cos\theta$$

Work is a scalar quantity with SI unit joule (J), where 1 J = 1 N m.

• Positive work — force has a component along displacement (e.g., gravity on a falling body).
• Negative work — force opposes displacement (e.g., friction).
• Zero work — force is perpendicular to displacement (e.g., centripetal force, or carrying a load horizontally).

Work Done by a Variable Force

When the force varies with position, work is the area under the force–displacement graph:

$$W = \int_{x_1}^{x_2} F\,dx$$

Kinetic Energy

The kinetic energy of a body of mass m moving with speed v is the energy it possesses by virtue of its motion:

$$K = \frac{1}{2}mv^2 = \frac{p^2}{2m}$$

Work-Energy Theorem

The work done by the net force acting on a body equals the change in its kinetic energy:

$$W_{net} = K_f - K_i = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$$

This theorem holds for both constant and variable forces and is a powerful shortcut that avoids dealing with acceleration directly.

Tip: The work-energy theorem uses the net work. If multiple forces act, add their individual works (with signs) before equating to ΔK.

CBSE/HBSE trap: A man holding a heavy suitcase stationary or walking horizontally with it does no work against gravity (θ = 90° or no displacement), even though he feels tired.

Potential Energy, Conservation of Energy and Power

Potential Energy

Potential energy is the energy possessed by a body by virtue of its position or configuration.

Gravitational potential energy near the Earth's surface:

$$U = mgh$$

Elastic potential energy of a spring stretched/compressed by x (spring constant k):

$$U = \frac{1}{2}kx^2$$

The restoring force of a spring is F = −kx (Hooke's law), and the work done to stretch it is stored as elastic PE.

Conservative and Non-Conservative Forces

• A conservative force (gravity, spring force) does work independent of path; total work in a closed loop is zero.
• A non-conservative force (friction, air resistance) depends on path and dissipates mechanical energy as heat.

Conservation of Mechanical Energy

For a body under only conservative forces, the total mechanical energy remains constant:

$$K + U = \text{constant} \quad\Rightarrow\quad \frac{1}{2}mv^2 + mgh = \text{constant}$$

For a freely falling body, as it descends, PE converts to KE while the sum stays fixed.

Power

Power is the rate of doing work or transferring energy:

$$P = \frac{W}{t} = \vec{F}\cdot\vec{v}$$

The SI unit is the watt (W), 1 W = 1 J s⁻¹. A commercial unit of energy is the kilowatt-hour (kWh), where 1 kWh = 3.6 × 10⁶ J.

Tip: Use P = Fv to find instantaneous power when force and velocity are known; use P = W/t for average power.

CBSE/HBSE trap: 1 kWh is a unit of energy, not power. Mixing up watt (power) and kWh (energy) is a common error.

Collisions in One and Two Dimensions

What is a Collision?

A collision is a short-duration interaction between bodies during which strong forces act and momentum is exchanged. In all collisions, the total linear momentum is conserved (no external force during the brief impact).

Elastic vs Inelastic Collisions

Elastic Collision in One Dimension

For masses m₁ (velocity u₁) and m₂ (velocity u₂), applying conservation of momentum and kinetic energy gives the final velocities:

$$v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1 + \frac{2m_2}{m_1 + m_2}u_2$$

$$v_2 = \frac{2m_1}{m_1 + m_2}u_1 + \frac{m_2 - m_1}{m_1 + m_2}u_2$$

Special cases (u₂ = 0):

• Equal masses: velocities are simply exchanged (v₁ = 0, v₂ = u₁).
• Heavy hits light (m₁ ≫ m₂): v₁ ≈ u₁, v₂ ≈ 2u₁.
• Light hits heavy (m₁ ≪ m₂): v₁ ≈ −u₁ (bounces back), v₂ ≈ 0.

Perfectly Inelastic Collision

The bodies stick and move with a common velocity:

$$v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$

Here kinetic energy is lost (converted to heat, sound, deformation), but momentum is conserved.

Coefficient of Restitution

$$e = \frac{\text{relative velocity of separation}}{\text{relative velocity of approach}}$$

For a perfectly elastic collision e = 1; for a perfectly inelastic collision e = 0.

Collisions in Two Dimensions

When the collision is oblique, momentum is conserved separately along two perpendicular axes (x and y). The four unknowns (two speeds and two angles) require resolving momentum in both directions plus the energy condition for elastic cases.

CBSE/HBSE trap: Momentum is conserved in every collision, but kinetic energy is conserved only in elastic collisions. Do not apply KE conservation to inelastic problems.

Frequently asked questions

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Yes — the Work, Energy and Power notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Work, Energy and Power notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Work, Energy and Power chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Work, Energy and Power.