Introduction
Kinetic energy is the energy of motion, defined as K = 1/2mv2. The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy. This theorem provides a scalar approach to mechanics problems, often simpler than vector force analysis.
Definition of Kinetic Energy
Kinetic energy K = 1/2mv2 where m is mass and v is speed. It is a scalar quantity, always non-negative. Kinetic energy depends on the reference frame (velocity is frame-dependent). Units: Joules (J). For a system of particles, total kinetic energy is the sum of individual kinetic energies: K_total = Σ 1/2m_i v_i2.
Work-Energy Theorem
The net work done by all forces acting on an object equals the change in its kinetic energy: W_net = ΔK = K_final - K_initial = 1/2mv2 - 1/2mv02. This theorem is derived from Newton's Second Law integrated over displacement. It applies to both constant and variable forces, to conservative and non-conservative forces. Positive work increases kinetic energy; negative work decreases it.
\nIntroduction
Kinetic energy is the energy of motion, defined as K = 1/2mv2. The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy. This theorem provides a scalar approach to mechanics problems, often simpler than vector force analysis.
Definition of Kinetic Energy
Kinetic energy K = 1/2mv2 where m is mass and v is speed. It is a scalar quantity, always non-negative. Kinetic energy depends on the reference frame (velocity is frame-dependent). Units: Joules (J). For a system of particles, total kinetic energy is the sum of individual kinetic energies: K_total = Σ 1/2m_i v_i2.
Work-Energy Theorem
The net work done by all forces acting on an object equals the change in its kinetic energy: W_net = ΔK = K_final - K_initial = 1/2mv2 - 1/2mv02. This theorem is derived from Newton's Second Law integrated over displacement. It applies to both constant and variable forces, to conservative and non-conservative forces. Positive work increases kinetic energy; negative work decreases it.
\nDerivation from Newton's Laws
Starting with F = ma = m(dv/dt), multiply both sides by v = dx/dt: Fv = mv(dv/dt). Recognizing v(dv/dt) = d(1/2v2)/dt, we get F(dx/dt) = d(1/2mv2)/dt. Multiply by dt and integrate: ∫F dx = Δ(1/2mv2). The left side is work, right side is change in kinetic energy. This derivation shows the deep connection between Newtonian mechanics and energy methods.
Applications
Applications include: finding final speed given forces and displacement (without time); determining stopping distance for vehicles; analyzing motion with position-dependent forces; calculating work done by friction over a path. Example: A block sliding down a rough incline - find speed at bottom using W_gravity + W_friction = ΔK, avoiding time-dependent kinematics.
Limitations and Extensions
The Work-Energy Theorem applies to a single particle or to rigid bodies where all parts have the same velocity. For deformable bodies or systems with internal motion, must account for internal work and different velocities of parts. Extended to systems: W_ext + W_int,non-cons = ΔK + ΔU + ΔE_int, accounting for internal energy changes.
\nSolved Example: Car Stopping Distance
A 1200 kg car moving at 30 m/s brakes with constant force. Stopping distance is 60 m. Find braking force and stopping time using work-energy theorem. Solution: Initial KE = 1/2mv02 = 1/2 × 1200 × 900 = 540,000 J. Final KE = 0. Work done by friction: W = -F × 60. By work-energy: -60F = 0 - 540,000 → F = 9000 N (braking force). Now using kinematics to find time: a = F/m = -9000/1200 = -7.5 m/s2. Using v = v0 + at: 0 = 30 - 7.5t → t = 4 s. Verify: distance = v0t + 1/2at2 = 30×4 + 1/2(-7.5)(16) = 120 - 60 = 60 m ✓. Work-energy method avoided needing time information initially.
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