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

Concept: The Work-Energy Theorem establishes a direct relationship between force and motion. It states that the total work done by the net force on a particle equals the change in its kinetic energy. Mathematically, \[ W = \Delta K \] where, \[ K = \frac{1}{2}mv^2 \] This theorem connects Newton’s Laws of Motion with energy principles. 
Mathematical Proof: Consider a particle of mass \( m \) moving under a constant force \( F \). From Newton’s Second Law: \[ F = ma \] We know that acceleration \( a \) can be written as: \[ a = \frac{dv}{dt} \] Using the chain rule: \[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} \] Since \( \frac{dx}{dt} = v \), we get: \[ a = v \frac{dv}{dx} \] Substitute into \( F = ma \): \[ F = m v \frac{dv}{dx} \] Rearranging: \[ F \, dx = m v \, dv \] Integrating both sides between initial and final states: \[ \int_{x_1}^{x_2} F \, dx = \int_{v_1}^{v_2} m v \, dv \] Left side represents total work done \( W \): \[ W = m \int_{v_1}^{v_2} v \, dv \] \[ W = m \left[ \frac{v^2}{2} \right]_{v_1}^{v_2} \] \[ W = \frac{1}{2}m v_2^2 - \frac{1}{2}m v_1^2 \] \[ W = K_2 - K_1 \] \[ \boxed{W = \Delta K} \] Thus, the net work done on a particle is equal to the change in its kinetic energy.