A 64-kg woman stands on frictionless ice. She kicks a 0.10-kg stone backwards with her feet so that the stone acquires a velocity of 1.1m/s. The velocity (in m/s) acquired by the woman is :

Options

A. 1.1m/s forward
B. 0.0017m/s backward
C. 0.0017m/s forward
D. 1.1m/s backward

Explanation

To solve the problem, we will use the principle of conservation of momentum. According to this principle, the total momentum before an event must equal the total momentum after the event, provided no external forces act on the system.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Mass of the woman, \( m_w = 64 \, \text{kg} \)
- Mass of the stone, \( m_s = 0.10 \, \text{kg} \)
- Velocity of the stone after being kicked, \( v_s = 1.1 \, \text{m/s} \)

2. **Understand the Initial Momentum:**
- Initially, both the woman and the stone are at rest, so the total initial momentum is:
\[ p_{\text{initial}} = 0 \]

3. **Set Up the Momentum Conservation Equation:**
- After the woman kicks the stone, the momentum of the woman and the stone must equal the initial momentum:
\[ m_w \cdot v_w + m_s \cdot v_s = 0 \]
- Here, \( v_w \) is the velocity of the woman after kicking the stone. The direction of the woman's velocity will be opposite to that of the stone.

4. **Rearranging the Equation:**
- Rearranging the equation gives:
\[ m_w \cdot v_w = - m_s \cdot v_s \]
- Therefore, we can express the velocity of the woman as:
\[ v_w = -\frac{m_s \cdot v_s}{m_w} \]

5. **Substituting the Values:**
- Substitute the known values into the equation:
\[ v_w = -\frac{0.10 \, \text{kg} \cdot 1.1 \, \text{m/s}}{64 \, \text{kg}} \]
- Calculating this gives:
\[ v_w = -\frac{0.11}{64} \approx -0.00171875 \, \text{m/s} \]

6. **Final Result:**
- The negative sign indicates that the direction of the woman's velocity is opposite to that of the stone. Therefore, the magnitude of the velocity acquired by the woman is:
\[ v_w \approx 0.0017 \, \text{m/s} \, \text{(forward)} \]

### Final Answer:
The velocity acquired by the woman is approximately \( 0.0017 \, \text{m/s} \) forward.