While in Earth's orbit, an 80-kg astronaut carrying a 20-kg tool kit is initially drifting toward a stationary (to her) space shuttle at a speed of 2 m/s. If she throws the tool kit toward the shuttle with a speed of 6 m/s as seen from the shuttle, her final speed is

Options

A. 1 m/s toward the shuttle
B. 4 m/s toward the shuttle.
C. 1 m/s away from the shuttle.
D. 6 m/s away from the shuttle.

Answer

To solve the problem, we need to apply the principle of conservation of momentum. The total momentum before the astronaut throws the toolkit must equal the total momentum after she throws it.

### Step-by-Step Solution:

1. **Identify the Masses and Initial Velocities**:
- Mass of the astronaut (m_a) = 80 kg
- Mass of the toolkit (m_t) = 20 kg
- Initial speed of the astronaut and toolkit (v_initial) = 2 m/s (towards the shuttle)
- Speed of the toolkit as seen from the shuttle (v_t) = 6 m/s (towards the shuttle)

2. **Calculate the Initial Momentum**:
The initial momentum (p_initial) of the system (astronaut + toolkit) is given by:
\[ p_{initial} = (m_a + m_t) \cdot v_{initial} = (80 \, \text{kg} + 20 \, \text{kg}) \cdot 2 \, \text{m/s} = 100 \, \text{kg} \cdot 2 \, \text{m/s} = 200 \, \text{kg m/s} \]

3. **Determine the Velocity of the Toolkit After Being Thrown**:
Since the toolkit is thrown towards the shuttle, its speed relative to the astronaut (who is moving towards the shuttle) will be:
\[ v_{toolkit, final} = v_{initial} + v_t = 2 \, \text{m/s} + 6 \, \text{m/s} = 8 \, \text{m/s} \]
The toolkit moves at 8 m/s towards the shuttle.

4. **Calculate the Final Momentum**:
Let the final speed of the astronaut after throwing the toolkit be \( v_f \). The final momentum (p_final) can be expressed as:
\[ p_{final} = m_a \cdot v_f + m_t \cdot v_{toolkit, final} \]
Substituting the known values:
\[ p_{final} = 80 \, \text{kg} \cdot v_f + 20 \, \text{kg} \cdot 8 \, \text{m/s} \]
\[ p_{final} = 80 \, \text{kg} \cdot v_f + 160 \, \text{kg m/s} \]

5. **Set Initial Momentum Equal to Final Momentum**:
According to the conservation of momentum:
\[ p_{initial} = p_{final} \]
Therefore:
\[ 200 \, \text{kg m/s} = 80 \, \text{kg} \cdot v_f + 160 \, \text{kg m/s} \]

6. **Solve for the Final Velocity of the Astronaut**:
Rearranging the equation:
\[ 80 \, \text{kg} \cdot v_f = 200 \, \text{kg m/s} - 160 \, \text{kg m/s} \]
\[ 80 \, \text{kg} \cdot v_f = 40 \, \text{kg m/s} \]
\[ v_f = \frac{40 \, \text{kg m/s}}{80 \, \text{kg}} = 0.5 \, \text{m/s} \]

### Final Answer:
The final speed of the astronaut after throwing the toolkit is **0.5 m/s** towards the shuttle.