A rocket of mass `m_0` has attained a speed equal to its exhaust speed and that time the mass of the rocket is `m`. Then the ratio `m_0/m` is (neglect gravity)

Options

A. (a) `2.718`
B. (b) `7.8`
C. (c) `3.14`
D. (d) `4`

Explanation

To solve the problem, we will use the rocket equation derived from the conservation of momentum. The steps are as follows:

### Step 1: Understand the Problem
We have a rocket with an initial mass \( m_0 \) and it has attained a speed equal to its exhaust speed \( V_{ex} \) at a certain time when its mass is \( m \). We need to find the ratio \( \frac{m_0}{m} \).

### Step 2: Write the Rocket Equation
The velocity of the rocket can be expressed using the Tsiolkovsky rocket equation:
\[ V = V_{ex} \ln\left(\frac{m_0}{m}\right) \]
where:
- \( V \) is the final speed of the rocket,
- \( V_{ex} \) is the exhaust speed,
- \( m_0 \) is the initial mass of the rocket,
- \( m \) is the mass of the rocket at the time it has attained speed \( V \).

### Step 3: Set the Speed Equal to Exhaust Speed
According to the problem, the speed of the rocket \( V \) is equal to its exhaust speed \( V_{ex} \):
\[ V = V_{ex} \]

### Step 4: Substitute into the Rocket Equation
Substituting \( V \) into the rocket equation gives:
\[ V_{ex} = V_{ex} \ln\left(\frac{m_0}{m}\right) \]

### Step 5: Simplify the Equation
We can divide both sides by \( V_{ex} \) (assuming \( V_{ex} \neq 0 \)):
\[ 1 = \ln\left(\frac{m_0}{m}\right) \]

### Step 6: Exponentiate Both Sides
To eliminate the logarithm, we exponentiate both sides:
\[ e^1 = \frac{m_0}{m} \]
Thus, we have:
\[ \frac{m_0}{m} = e \]

### Step 7: Final Result
The value of \( e \) is approximately \( 2.718 \). Therefore, the ratio \( \frac{m_0}{m} \) is:
\[ \frac{m_0}{m} \approx 2.718 \]

### Conclusion
Thus, the final answer is:
\[ \frac{m_0}{m} = 2.718 \]

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