Question

An astronaut with a mass of 70.0 kg is outside a space capsule when the tether line breaks. To return to the capsule, the astronaut throws a 2.0 kg wrench away from the capsule at a speed of 14 m/s. At what speed does the astronaut move toward the capsule?
a
3.5 m/s
b
7.0 m/s
c
5.0 m/s
d
0.4 m/s

Answer 1

To solve this problem, we can use the principle of conservation of momentum. The total momentum of the system (astronaut + wrench) before and after the throw must be the same since there are no external forces acting on them.

1. **Initial State:**
Before the wrench is thrown, both the astronaut and wrench are at rest:
\[
\text{Initial momentum} = 0
\]

2. **Final State:**
After the wrench is thrown, we'll designate the speed of the astronaut moving towards the capsule as \( v_a \) and the velocity of the wrench as \( v_w \) (14 m/s away from the capsule).

- Mass of the astronaut \( m_a = 70.0 \) kg
- Mass of the wrench \( m_w = 2.0 \) kg
- Velocity of the wrench \( v_w = 14 \) m/s

Using the conservation of momentum:
\[
0 = m_a \cdot v_a + m_w \cdot (-v_w)
\]
(Note: The wrench's velocity is negative because it is thrown away from the capsule.)

Plugging in the values we have:
\[
0 = 70.0 \cdot v_a + 2.0 \cdot (-14)
\]

This simplifies to:
\[
0 = 70.0 \cdot v_a - 28
\]

Solving for \( v_a \):
\[
70.0 \cdot v_a = 28
\]
\[
v_a = \frac{28}{70.0} = 0.4 \text{ m/s}
\]

Thus, the speed at which the astronaut moves towards the capsule is **0.4 m/s**.

The correct answer is **d: 0.4 m/s**.