Energy Conservation

Kinetic energy is $KE = \tfrac{1}{2}mv^2$; gravitational potential energy is $PE = mgh$; spring energy is $\tfrac{1}{2}kx^2$. With only conservative forces acting, $KE_i + PE_i = KE_f + PE_f$ Conservative forces (gravity, springs) are path-independent and store recoverable energy; friction and air resistance are non-conservative and drain mechanical energy into heat, so $\Delta E_{mech} = W_{friction}$. Energy methods beat force methods whenever the path is curved or complicated.

Power

Power is the rate of energy transfer: $P = \frac{W}{t} = Fv$ measured in watts. Machines and ramps trade force for distance, but never reduce the work required.

Key Takeaways

• Only the force component along the displacement does work.
• Conservation of energy solves ramp, pendulum, and roller-coaster problems fastest.
• Friction converts mechanical energy to heat; account for it as lost energy.
• $P = Fv$ handles constant-velocity engine and motor questions.