Find the mass of the rocket as a function of time, if it moves with a constant acceleration `a`, in absence of external forces. The gas escapes with a constant velocity `u` relative to the rocket and its mass initially was `m_0`.

Explanation

To find the mass of the rocket as a function of time under the given conditions, we can follow these steps:

### Step 1: Understand the forces acting on the rocket
In the absence of external forces, the only force acting on the rocket is the thrust force generated by the escaping gas. The thrust force can be expressed in terms of the rate of change of mass and the velocity of the gas.

### Step 2: Write the equation of motion
According to Newton's second law, the net force acting on the rocket is equal to the mass of the rocket multiplied by its acceleration:
\[ F_{\text{net}} = m \cdot a \]
The thrust force can also be expressed as:
\[ F_{\text{thrust}} = -V_r \frac{dm}{dt} \]
where \( V_r \) is the relative velocity of the gas escaping from the rocket and \( \frac{dm}{dt} \) is the rate of change of mass.

### Step 3: Set up the equation
Equating the two expressions for force:
\[ m \cdot a = -V_r \frac{dm}{dt} \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ a \cdot dt = -V_r \frac{dm}{m} \]

### Step 5: Integrate both sides
Integrating both sides, we have:
\[ \int a \, dt = -V_r \int \frac{dm}{m} \]
The limits for \( t \) are from 0 to \( t \) and for \( m \) from \( m_0 \) to \( m \):
\[ a \cdot t = -V_r \left( \ln(m) - \ln(m_0) \right) \]

### Step 6: Simplify the equation
This simplifies to:
\[ a \cdot t = -V_r \ln\left(\frac{m}{m_0}\right) \]

### Step 7: Solve for mass \( m \)
Rearranging gives:
\[ \ln\left(\frac{m}{m_0}\right) = -\frac{a \cdot t}{V_r} \]
Exponentiating both sides results in:
\[ \frac{m}{m_0} = e^{-\frac{a \cdot t}{V_r}} \]
Thus, we can express the mass as:
\[ m = m_0 \cdot e^{-\frac{a \cdot t}{V_r}} \]

### Step 8: Substitute \( V_r \) with \( u \)
Since the problem states that the gas escapes with a constant velocity \( u \) relative to the rocket, we can replace \( V_r \) with \( u \):
\[ m(t) = m_0 \cdot e^{-\frac{a \cdot t}{u}} \]

### Final Answer
The mass of the rocket as a function of time is:
\[ m(t) = m_0 \cdot e^{-\frac{a \cdot t}{u}} \]

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