Work-energy Theorem

The work-energy theorem states that the total work done by all forces on an object equals the change in kinetic energy of that object. This relationship connects force, displacement, and velocity through the equation W = ½m(v₂² - v₁²), where m is mass and v₁ and v₂ are initial and final velocities. The theorem provides a powerful tool for analyzing motion when acceleration varies.

Work is positive when an applied force increases an object's velocity and kinetic energy. Work is zero when a force causes no change in velocity or kinetic energy. Work is negative when a net force opposes motion, decreasing kinetic energy. Understanding positive, negative, and zero work helps predict how forces affect an object's motion and energy state.

Starting with Newton's second law (F = ma), you substitute force and displacement into the work formula. Using the third equation of motion to relate velocity and acceleration, you derive the relationship W = ½m(v₂² - v₁²). This mathematical derivation shows how work done by forces directly produces changes in kinetic energy.

First, identify given quantities and draw a free-body diagram showing all forces. Calculate work done by each known force, tracking signs carefully. Sum total work from all forces. Apply the work-energy theorem, equating total work to change in kinetic energy. Substitute known values and solve for unknowns. Finally, evaluate your answer for reasonableness.

According to Newton's second law, the net force determines how an object's motion changes. Therefore, only the total work done by all forces affects kinetic energy. Considering work and energy for variable forces requires summing contributions from every force acting on the object to accurately predict energy changes.

The work-energy theorem is especially valuable when acceleration is not constant. Instead of tracking acceleration throughout motion, you only need initial and final velocities to find work done. This makes the theorem ideal for analyzing complex motion scenarios where forces vary, providing a simpler alternative to kinematic equations.

If total work done is zero, the object's kinetic energy remains unchanged, meaning it travels at constant speed. This occurs when forces are balanced or when applied forces do no net work on the object. Zero total work indicates the object maintains its current velocity without acceleration or deceleration.