Question

State the Work-Energy Theorem and provide its mathematical proof.

Hint:

Work–Energy Theorem provides an alternative method to solve motion problems without directly using kinematic equations. If work done is known, velocity can be found using energy relations.

Answer 1

Work–Energy Theorem:
The Work–Energy Theorem states that the work done by a net force acting on a body is equal to the change in its kinetic energy.

Mathematically,
W = ΔK
or
W = (1/2)mv² − (1/2)mu²

where m is the mass of the body, u is the initial velocity, and v is the final velocity.

Mathematical Proof:

Step 1: Consider a body of mass m
Let a constant force F act on a body of mass m. Due to this force, the body accelerates from initial velocity u to final velocity v in a distance s.

From Newton’s Second Law,
F = ma

Step 2: Expression for Work Done
Work done by the force,
W = F × s

Substituting F = ma,
W = ma × s

Step 3: Use Equation of Motion
From the third equation of motion,
v² = u² + 2as

Rearranging,
2as = v² − u²

⇒ as = (v² − u²)/2

Step 4: Substitute in Work Expression
W = m(as)
W = m[(v² − u²)/2]

W = (1/2)m(v² − u²)

Step 5: Rearranging
W = (1/2)mv² − (1/2)mu²

Since kinetic energy (K) = (1/2)mv²,

W = Final kinetic energy − Initial kinetic energy
W = ΔK

Conclusion:
Thus, the work done by the net force on a body is equal to the change in its kinetic energy. This proves the Work–Energy Theorem.