A 70.0 kg astronaut pushes to the left on a spacecraft with a force `vec(F)` in "gravity-free" space. The spacecraft has a total mass of `1.0xx10^(4)` kg. During the push, the astronaut accelerates to the right with an acceleration of `0.36" m"//"s"^(2)`.
Which one of the following statements concerning this situation is true ?

Options

A. The spacecraft does not move, but the astronaut moves to the right with a constant speed.
B. The astronaut stops moving after he stops pushing on the spacecraft.
C. The force exerted on the astronaut is larger than the force exerted on the spacecraft.
D. The velocity of the astronaut increases while he is pushing on the spacecraft.

Answer

To solve the problem, we will analyze the forces acting on the astronaut and the spacecraft, and use Newton's laws of motion to determine which statement is true.

### Step 1: Identify the forces acting on the astronaut and the spacecraft.
- The astronaut exerts a force \( \vec{F} \) to the left on the spacecraft.
- According to Newton's third law of motion, the spacecraft exerts an equal and opposite force on the astronaut, which is directed to the right.

### Step 2: Calculate the force exerted by the astronaut.
- The astronaut has a mass of \( m_a = 70.0 \, \text{kg} \) and accelerates to the right with an acceleration \( a = 0.36 \, \text{m/s}^2 \).
- Using Newton's second law, the force exerted by the astronaut can be calculated as:
\[ F_a = m_a \cdot a = 70.0 \, \text{kg} \cdot 0.36 \, \text{m/s}^2 = 25.2 \, \text{N} \]

### Step 3: Analyze the motion of the spacecraft.
- The spacecraft has a mass of \( m_s = 1.0 \times 10^4 \, \text{kg} \).
- The force exerted on the spacecraft by the astronaut is \( F_s = 25.2 \, \text{N} \) (to the left).
- The acceleration of the spacecraft can be calculated using Newton's second law:
\[ a_s = \frac{F_s}{m_s} = \frac{25.2 \, \text{N}}{1.0 \times 10^4 \, \text{kg}} = 0.00252 \, \text{m/s}^2 \]
- This indicates that the spacecraft does move, but with a very small acceleration to the left.

### Step 4: Evaluate the statements.
1. **The spacecraft does not move but the astronaut moves to the right with a constant speed.**
- **False.** The spacecraft does move, albeit slowly, and the astronaut accelerates to the right.

2. **The astronaut stops moving after he stops pushing on the spacecraft.**
- **False.** Once the astronaut pushes, he gains velocity and will continue moving at a constant speed after he stops pushing, due to inertia.

3. **The force exerted on the astronaut is larger than the force exerted on the spacecraft, equal and opposite.**
- **False.** According to Newton's third law, the forces are equal and opposite.

4. **The velocity of the astronaut increases while he is pushing on the spacecraft.**
- **True.** The astronaut accelerates to the right while pushing on the spacecraft.

### Conclusion:
The correct statement is: **The velocity of the astronaut increases while he is pushing on the spacecraft.**

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