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?

Question

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.