MAXIMUM Velocity of a Block Pushed by a Spring (With Friction) | Mechanical Energy | Physics

Use the conservation of mechanical energy as well as Hooke's Law to determine the MAXIMUM velocity of a block as it is pushed forward by a compressed spring across a rough surface. In this problem, potential energy, initially stored in the spring is transferred into a block of mass m by the elastic force, causing the block to accelerate forward. As the block is pushed across the rough surface the friction force acts as a nonconservative force on the block and removes mechanical energy from the system, turning mechanical energy into thermal energy. Solve for the work done by the spring, the work done by friction and the velocity off the block as it passes through equilibrium. The misconception in this problem is that the maximum velocity of the block does not occur at equilibrium. This is because after a certain point the force forward by the spring, even though it is still acting on the block, is less than the friction force which is working to slow the block. Taking the derivative of kinetic energy with respect to position and setting that value equal to zero shows not only where the block is going the fastest, but it also reduces to Newton's second law showing that at the position of maximum velocity the friction force equals the spring force.