The total work can be split up into two parts:
$$W_{net} = W_{conservative}+W_{non-conservative}.$$
With the conservative part you can associate a potential energy:
$$W_{conservative}=-\Delta PE$$
(this is in fact the definition of a conservative force) so that the Work-Energy theorem becomes
$$W_{non-conservative}=\Delta KE + \Delta PE = \Delta E.$$
This is another way of writing the Work-Energy theorem and in my mind it's a little bit clearer. Restated, the work done by non-conservative forces is equal to the overall change in energy of the system.
For example, work done by friction is negative, so it dissipates energy away from a system.
On the other hand, gravity is a conservative force. Imagine the motion of a falling ball. Unless something doing work on the ball to slow it down (for example, air) the ball will speed up as it falls. In this case, the equation
$$W_{gravity} = -\Delta PE = \Delta KE$$
is equivalent to that statement. (As the potential energy becomes more negative, kinetic energy becomes more positive.)
Both your equations are valid as long as you're dealing with conservative forces. They - pretty much by definition - express conservation of mechanical energy. There are other kinds of energy as well and in most realistic situations you need to take them into account as well. The total energy is always conserved and for conservative forces in classical mechanics, the only relevant types of energy (the ones that can change) are potential and kinetic energy. So the sum of those is conserved.
In the situation of the car the forces are not conservative and energy is lost through e.g. heat from friction in the engine. The potential energy that has been decreased is the chemical bond energy of the fuel. Fuel is being 'burned', i.e. bonds are being broken, and the energy that is released by this process is used to perform work. However, not all the energy is put to good use, there's losses from the system heating up (and that thermal energy is mostly released into the environment). There's also losses from friction of the wheels etc. but the total energy, minus the losses, is conserved.
Best Answer
The state of constant motion along a straight path or being at rest defines a state of the system known as inertia. Force is some agency that disturbs the state of inertia. Force causes acceleration (precisely, a change in momentum). Newton's second law of motion states that
$$\vec{F}=\frac{d\vec{p}}{dt}$$
If the body is moving at a constant velocity, then there is no change in momentum and so
$$\frac{d\vec{p}}{dt}=0\implies\vec{F}=0$$
Hence constant motion means no external unbalanced force is acting on the body. Since there is no force, there is no work done (Since $W=\int\vec{F}\cdot d\vec{s}$).
Hence no work is necessary to be done on a body to maintain it at a constant speed, because no force is required for a body to move with constant velocity. There is displacement of course. The fact is that the displacement that happens here during a constant motion here is not the consequence of the force. The displacement term appearing in the equation of work:
$$\text{Work done}= \text{Force} \times \text{displacement along the direction of applied force}$$
is the displacement as a consequence of the applied force. The dot product between force and displacement (picks up the component of displacement in the direction of the force) vanishes if the force and displacement are independent of each other (or orthogonal to each other in some sense). Here the displacement happens not due to any force. So, the component of force that caused the displacement is zero. Hence there is no work done.
Then by the work energy theorem, there is no change in the kinetic energy of the body. Hence the conservation of energy. This makes sense. No work is done on the body as there is no force to do that. The body is under constant motion, which means momentum is conserved. That is, $p=\text{constant}\implies \Delta p=0$ and so $K.E=p^2/2m=\text{constant}\implies\Delta K.E=(\Delta p)^2/2m=0$.
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