Now we do $-ve$ work on the system while stopping it so the work done by the system is $+ve$. Is this part right?
Not always; you've stumbled upon a subtle point in the definition of work in mechanics, which is rarely discussed. Generally, if body $A$ has done work $W$ on body B, it does not follow that the body $B$ has done work $-W$ on the body $A$. This is true only if the two material points under action of the mutual forces have moved with the same velocity.
Explanation:
Definition of rate of work: When body S acts with force $\mathbf F$ on body $\mathbf{B}$ that has velocity $\mathbf v_B$ (more accurately, it is the velocity of the mass point which experiences the force), the rate of work being done by S on $\mathbf{B}$ is defined to be
$$
\mathbf F\cdot\mathbf v_B.
$$
Notice that the velocity is that of the receiver, not that of the body the force is due to. So if I scratch my desk while the scratched portion remains at rest, no work has been done on the desk.
Thus, the definition of work is based on:
- the force due to the giving body;
- the velocity of the receiving body (its mass point the force acts on).
What about the work done on the body $S$? This is, per definition,
$$
-\mathbf F\cdot\mathbf v_S.
$$
The forces have the same magnitude and opposite signs (due to Newton's 3rd law), but there is no general relation between the two velocities of mass points where the forces are acting. If $\mathbf v_S$ is not equal to $\mathbf v_B$ during the whole process, it is possible the two works done on the bodies will not have the same magnitude.
⁂
Let's look at some specific cases.
Case 1. If a massive block $\mathbf B$ is brought to rest by another moving body $S$ with no sliding friction occurring (if the mass points of bodies experiencing the mutual forces always move with the same velocity), the velocities $\mathbf v_B,\mathbf v_S$ are the same and the two works have the same magnitude and opposite sign.
This is the case, for example, when the block is stopped in its motion by a spring mounted on a wall, or a person stops it gradually by hand. The kinetic energy of the block $E_k$ decreases to zero and equal amount of energy is added through work to the total energy of the stopping body. No heat transfer and no change in temperature need to occur, if there is no sliding friction and no temperature differences beforehand.
Case 2. If the block is stopped by forces of sliding friction - say, due to the ground - the description in terms of work is different. The mass point where the force due to ground acts on the block $\mathbf B$ is part of the block and is moving. Therefore the ground is doing work on the block (and from the reference frame of the ground, this work is negative). However, since the ground is not moving at all, the block does zero work on the ground!
This may look like violation of energy conservation, because block is slowing down without ground receiving energy, for there is no work received.
But it is only violation of mechanical energy conservation, which is fine and occurs daily. Total energy may still remain conserved, because it includes also internal energy of the block and internal energy of the ground, which change during the process.
As the block slows down, its kinetic energy transforms into different form of energy: internal energy of both the ground and the block. This happens with greatest intensity in the two faces that are in mutual mechanical contact.
The faces get warmer and for the rest of the system, they act as heat reservoirs.
The energy is transferred via heat both upwards into the block and downwards into the ground.
Further , isn't work done always equal to change in kinetic energy (by the work energy theorem) ?
Only if the only energy that changes is kinetic energy. Generally, the work-energy theorem includes other energies. Kinetic energy can change into potential energy (in gravity field, in a spring) or into internal energy (inside matter, may manifest as increase in temperature or other change of thermodynamic state).
I think the first law of thermodynamics could be restated as
\begin{equation}
\Delta U_S + \Delta U_{\Omega\setminus S} = 0
\end{equation}
i.e.
\begin{equation}
U_S+U_{\Omega\setminus S}=\text{constant}
\end{equation}
and this would clarify (to me) the relationship between the two concepts.
What you describe is a general law of conservation of energy; it assumes the rest of the universe (surroundings) can be ascribed energy and that the sum is conserved. The problem with this law is that we have no means of controlling the surroundings by definition, so it is (rather well-working) leap of faith.
The First law of thermodynamics is a little bit more restricted and more experimentally grounded. It concerns itself just with the system $S$, the surroundings are left unaccounted for.
It has more formulations which are more or less are equivalent. One of them is
Effect of heat supplied to the system on its state is equivalent to effect of certain equivalent amount of work supplied to it.$^{*}$ If both heat and work are measured in the same units (commonly Joules), a quantity characterizing the equilibrium state $X$, called internal energy, can be defined for all $X$. After the system is supplied heat $\Delta Q$ and work $\Delta W$, internal energy changes by
$$
\Delta U = \Delta Q + \Delta W.
$$
(end of the law).
The change of $X$ and $U$ does not necessarily determine values of $\Delta Q$, $\Delta W$; they depend on the way the process is executed. The first law only says whatever the process (reversible, irreversible), change in $U$ is given by sum of heat and work supplied.
$^*$ The work is to be done irreversibly in such a way as to mimic addition of heat, i.e. by stirring the fluid in the system. No amount of reversible work would make the system end up in the same state that addition of heat does (since addition of heat does not conserve system's entropy but reversible work does).
Best Answer
“Conserved” doesn’t mean “never changes”. It means “this stuff is real, and the only way you have less or more is if some is taken away or added”. You can then follow that additions or subtractions.
Since your cold bottle has less energy, the conservation law says that energy has not disappeared, it’s just gone somewhere. You can find it. You can figure out how it got there.