Even a simple act like pushing a toy truck across a carpet involves friction as well as kinetic energy.
Even a simple act like pushing a toy truck across a carpet involves friction as well as kinetic energy.

The work-energy theorem states that the change in kinetic energy of an object is equal to the work done on that object, but this equation is only valid for frictionless processes. Conservation of energy, in which the sum of the initial kinetic and potential energies is equal to the sum of the final kinetic and potential energy, is technically called the conservation of mechanical energy because it also assumes no friction or air resistance. In the real world, and often in challenging physics problems, friction plays an undeniable role, so it is important to understand how to incorporate friction into kinetic energy problems.

Find the Work Due to Friction

Find the normal force on the object. If the object is resting on a horizontal surface, the normal force is the same magnitude, or numerical value, as the force of gravity, F=mg, but in the opposite direction as the force of gravity. If the object is on a slanted surface, or other forces are acting on the object, you will have to take components, or directions, into account.

Multiply the normal force by the coefficient of kinetic friction.

Multiply that product by the distance the object moves to find the work done by friction.

Take the Work Due to Friction into Account

If you are using the work-energy theorem, add the work due to the force of friction to the same side of the equation as the change in kinetic energy.

If you are using the conservation of energy equation, add the work due to the force of friction to the same side of the equation as the initial kinetic and potential energies.

Now that you have the complete equation, solve for the variable you want.