3. (6 points) In a vertical stationary cylindrical vessel closed by a piston of mass $M$, an ideal gas is located. The gas is heated, and the piston moves from rest with a constant acceleration $a$. During the heating time, the piston exits the vessel and then continues to move vertically upward in free space. The average power of the heater is $P$. The molar heat capacity of the gas at constant volume is $c_{\mu \nu}$. The acceleration due to gravity is $g$. The heat capacities of the vessel and the piston are negligible. There is a vacuum outside the piston.

Determine the maximum height the piston rises above the edge of the vessel.
Possible solution. Since the piston moves with uniform acceleration, the process is isobaric. According to the first law of thermodynamics,

$$
Q=p \Delta V+v c_{\mu V} \Delta T
$$

Here $p$ - is the pressure of the gas, $V$ - is its volume, $v$ - is the amount of substance of the gas.

According to the Clapeyron-Mendeleev equation,

$$
p \Delta V=v R \Delta T
$$

By the theorem of kinetic energy change,

$$
p \Delta V=\frac{M(a \tau)^{2}}{2}
$$

Thus,

$$
v \Delta T=\frac{M(a \tau)^{2}}{2 R}
$$

The amount of heat transferred:

$$
Q=\frac{M(a \tau)^{2}}{2}+\frac{c M(a \tau)^{2}}{2 R}=\frac{M(a \tau)^{2}}{2}\left(1+\frac{c_{\mu V}}{R}\right) .
$$

During the time of movement in the cylinder, the piston acquired a speed

$$
V=a \tau
$$

Then

$$
P=\frac{Q}{\tau}=\frac{M a V}{2}\left(1+\frac{c_{\mu V}}{R}\right) .
$$

Considering that

$$
V=g \tau^{\prime}
$$

in the final form we get

$$
\tau^{\prime}=\frac{2 P}{\operatorname{Mag}\left(1+\frac{C_{\mu V}}{R}\right)}
$$

Answer: $\tau^{\prime}=\frac{2 P}{\operatorname{Mag}\left(1+\frac{c_{\mu V}}{R}\right)}$.

## Evaluation Criteria

| Task | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for the first law of thermodynamics is written | $\mathbf{1}$ |
| The Clapeyron-Mendeleev equation is written | $\mathbf{1}$ |
| The theorem of kinetic energy change is written | $\mathbf{1}$ |
| The expression for the connection between power and piston speed is obtained |  |
| Necessary transformations are performed, the final answer is obtained | $\mathbf{2}$ |
| Total points | $\mathbf{6}$ |