Two fat astronauts each of mass 120 kg are travelling in a closed spaceship moving at a speed of 15 km/s in the outer space far removed from tall other material objects. The total mass of the spaceship and its contents including the astronauts is 660 kg. If the astronauts do slimming exercise and thereby reduce their masses to 90 kg each, with what velocity will spaceship move?

Explanation

To solve the problem, we will use the principle of conservation of momentum. The total momentum of a closed system remains constant if no external forces are acting on it.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- Mass of each astronaut before slimming: \( m_1 = 120 \, \text{kg} \)
- Total mass of the two astronauts: \( 2 \times m_1 = 240 \, \text{kg} \)
- Mass of the spaceship: \( m_s = 660 \, \text{kg} - 240 \, \text{kg} = 420 \, \text{kg} \)
- Initial speed of the spaceship: \( v_i = 15 \, \text{km/s} = 15000 \, \text{m/s} \)

2. **Calculate Initial Momentum:**
- Total initial mass of the system: \( M_i = m_s + 2 \times m_1 = 660 \, \text{kg} \)
- Initial momentum \( P_i \) of the system:
\[ P_i = M_i \times v_i = 660 \, \text{kg} \times 15000 \, \text{m/s} = 9900000 \, \text{kg m/s} \]

3. **Identify Final Conditions:**
- Mass of each astronaut after slimming: \( m_2 = 90 \, \text{kg} \)
- Total mass of the two astronauts after slimming: \( 2 \times m_2 = 180 \, \text{kg} \)
- Mass of the spaceship remains unchanged: \( m_s = 420 \, \text{kg} \)
- Total final mass of the system: \( M_f = m_s + 2 \times m_2 = 420 \, \text{kg} + 180 \, \text{kg} = 600 \, \text{kg} \)

4. **Apply Conservation of Momentum:**
- According to the conservation of momentum:
\[ P_i = P_f \]
- Therefore, we have:
\[ 9900000 \, \text{kg m/s} = M_f \times v_f \]
- Where \( v_f \) is the final velocity of the spaceship.

5. **Solve for Final Velocity:**
- Rearranging the equation gives:
\[ v_f = \frac{P_i}{M_f} = \frac{9900000 \, \text{kg m/s}}{600 \, \text{kg}} = 16500 \, \text{m/s} \]
- Converting back to km/s:
\[ v_f = 16.5 \, \text{km/s} \]

### Final Answer:
The velocity of the spaceship after the astronauts reduce their mass will be **16.5 km/s**.