What is the most simplified form of the law of conservation of energy for a block sliding to a stop?

The law of conservation of energy states that the total energy in a closed system remains constant. For a block sliding to a stop, we can analyze the energy transformations at play. Initially, the block has kinetic energy due to its motion, represented as ( \frac{1}{2} mv_i^2 ), where ( m ) is the mass of the block and ( v_i ) is its initial velocity.

As the block slides, it experiences non-conservative forces, such as friction, which do work on the system and typically convert kinetic energy into thermal energy, causing the block to lose energy. The work done by these non-conservative forces can be represented as ( W_{nc} ).

When the block comes to a stop, its final kinetic energy is zero, and the equation can be set up to represent this energy transfer:

[ \frac{1}{2} mv_i^2 + W_{nc} = 0 ]

This indicates that the initial kinetic energy plus the work done by non-conservative forces equals zero, meaning that all the kinetic energy has been transformed through work into other forms (like thermal energy due to friction).

This equation effectively captures the situation of the sliding block coming to rest