Work-Energy Theorem (Class 11)

Definition

The work-energy theorem states that the net work done by all the forces acting on a particle is equal to the change in its kinetic energy. When work is done on an object, it results in a change in the object’s kinetic energy.

Formula

W_net = ΔKE = KE_final – KE_initial

Where:

• W_net = net work done by all forces (Joule, J)
• KE_final = final kinetic energy (J)
• KE_initial = initial kinetic energy (J)
• KE = (1/2)mv² (m = mass, v = velocity)

Derivation

Consider a particle of mass m moving in a straight line under the action of a net force F.

From Newton’s second law:
F = m·a

If the particle moves a small distance dx under this force, the work done dW is:
dW = F·dx

Since acceleration a = dv/dt, and velocity v = dx/dt,
a = dv/dt = (dv/dx)·(dx/dt) = v·dv/dx

So,
F = m·a = m·v·dv/dx

Therefore,
dW = F·dx = m·v·dv/dx·dx = m·v·dv

Integrate both sides from initial velocity u to final velocity v:
∫(from u to v) m·v·dv = ∫(from x₁ to x₂) dW

m ∫(from u to v) v·dv = W_net

m [ (1/2) v² ] (from u to v) = W_net

m [ (1/2) v² – (1/2) u² ] = W_net

W_net = (1/2) m v² – (1/2) m u²

W_net = KE_final – KE_initial

This proves the work-energy theorem.

Example

A ball of mass 2 kg is initially at rest. A force acts on it and accelerates it to a velocity of 5 m/s.

• Initial velocity, u = 0
• Final velocity, v = 5 m/s
• Mass, m = 2 kg

Initial kinetic energy, KE_initial = (1/2) × 2 × (0)² = 0 J
Final kinetic energy, KE_final = (1/2) × 2 × (5)² = 25 J
Change in kinetic energy, ΔKE = 25 J – 0 J = 25 J

According to the work-energy theorem, the net work done on the ball is 25 J.

Key Points

• The theorem applies to all types of forces (conservative and non-conservative).
• If net work is positive, kinetic energy increases.
• If net work is negative, kinetic energy decreases.

Illustrative Image

Summary Table

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