Solution: Astronaut and Rocket Dynamics

1. Physics Principles

This problem is based on the Conservation of Linear Momentum in an isolated system.

• Inertial Frame: The frame moving with the floating astronaut (who is force-free) is an inertial frame.
• Conservation: In this frame, the initial total momentum of the spaceship is zero. Therefore, the final total momentum must also be zero.

2. Visual Analysis

3. Analysis of Options

Momentum Conservation:

$$ \vec{P}_{initial} = 0 $$ $$ \vec{P}_{final} = \vec{P}_{ship} + \vec{P}_{gas} = 0 $$ $$ \vec{P}_{ship} = – \vec{P}_{gas} $$

This implies that the magnitudes of their momenta are equal:

$$ |\vec{P}_{ship}| = |\vec{P}_{gas}| $$

Therefore, observation (d) is correct.

Kinetic Energy Comparison:

Kinetic energy can be expressed in terms of momentum ($P$) and mass ($m$):

$$ K = \frac{P^2}{2m} $$

We know $P_{ship} = P_{gas}$. Comparing their masses, the mass of ejected gases ($m_{gas}$) is typically much smaller than the mass of the remaining spaceship ($M_{ship}$).

$$ m_{gas} < M_{ship} $$

Since $K$ is inversely proportional to mass for the same momentum:

$$ \frac{P^2}{2M_{ship}} < \frac{P^2}{2m_{gas}} $$ $$ K_{ship} < K_{gas} $$

The spaceship has lesser kinetic energy than the ejected gases.

Therefore, observation (a) is correct.