Question

An ideal two-stage rocket has identical specific impulse and structural coefficient for its two stages. For an optimized rocket, the two stages have identical payload ratio as well. The payload is 2 tons and the initial mass of the rocket is 200 tons. The mass of the second stage of the rocket (including the final payload mass) is ___________ tons.

• A. 100
• B. 10
• C. 20
• D. 50

Hint:

In multi-stage rockets with identical payload ratios, each stage contributes equally to the total mass. By dividing the total mass of the rocket by the number of stages, we can easily find the mass of the second stage.

Answer 1

The Correct Option is C

The key information provided is that the two stages of the rocket have identical specific impulse and identical structural coefficient, which also means the identical payload ratio for both stages.

Step 1: Understanding the Payload Ratio
In an optimized multi-stage rocket, the payload ratio is the ratio of the payload mass to the total mass of the rocket stage.
Since both stages have identical payload ratios, the mass distribution between the stages will be proportional.

Let the total initial mass of the rocket be \( M_{{total}} = 200 \) tons.
The payload is given as 2 tons.

Step 2: Identifying the Mass Distribution
Let’s assume that \( M_1 \) is the mass of the first stage and \( M_2 \) is the mass of the second stage, including the payload.
We know that for a two-stage rocket with identical payload ratios:

\[ \frac{M_{{payload}}}{M_{{total}}} = \frac{M_{{payload, 1st stage}}}{M_1} = \frac{M_{{payload, 2nd stage}}}{M_2} \]

Where:
\( M_{{payload}} \) = 2 tons (total payload),
\( M_1 \) = mass of the first stage,
\( M_2 \) = mass of the second stage including the final payload.

Since both stages have identical payload ratios, each stage will contribute half of the total mass of the rocket.
Therefore, the mass of the second stage, including the payload, is:

\[ M_2 = \frac{200}{10} = 20 \, \text{tons} \]

Thus, the mass of the second stage (including the final payload) is 20 tons.

Therefore, the correct answer is:

\[ \boxed{(C) 20} \]