Question 8.2: Astronaut rescue In this example we will consider conservati...
Astronaut rescue
In this example we will consider conservation of momentum in an isolated system consisting of an astronaut and a wrench. An astronaut is floating in space 100 m from her ship when her safety cable becomes unlatched. She and the ship are motionless relative to each other. The astronaut’s mass (including space suit) is 100 kg; she has a 1.0 kg wrench and only a 20 minute air supply. Thinking back to her physics classes, she devises a plan to use conservation of momentum to get back to the ship safely by throwing the wrench away from her. In what direction should she throw the wrench? What is the magnitude of her recoil velocity if she throws the wrench at 10 m/s? Will her recoil velocity be great enough to get her back to the spacecraft before she runs out of air?
SET UP We choose a system consisting of the astronaut plus the wrench. No external forces act on this system, so the total momentum is conserved. We draw diagrams showing the situation just before and just after she throws the wrench; we point the x axis to the right (Figure 8.4). We use the subscripts i and f (initial and final) to label the situations before and after the wrench is thrown, and we use the subscripts A and W(astronaut and wrench) to identify the parts of the system.
SOLVE The astronaut needs to acquire a velocity toward the spaceship. Because the total momentum (of her and the wrench) is zero, she should throw the wrench directly away from the ship. All the vector quantities lie along the x axis, so we’re concerned only with x components. We write an equation expressing the equality of the initial and final values of the total x component of momentum:
m_A(\nu _{A,i,x})+m_W(\nu _{W,i,x})=m_A(\nu _{A,f,x})+m_W(\nu _{W,f,x})
In this case, the initial velocity of each object is zero, so the left side of the equation is zero; that is, m_A(\nu _{A,i,x})+m_W(\nu _{W,i,x})=0. Solving for \nu_{A,f, x}, we get the astronaut’s x component of velocity after she throws the wrench:
\upsilon _{A,f,x}=\frac{-m_W(\nu _{W,f,x})}{m_A}=\frac{-(1.0 kg)(10.0 m/s)}{(100 kg)} =0.10 m/s.
The negative sign indicates that the astronaut is moving toward the ship, opposite our chosen +x direction.
To find the total time required for the astronaut to travel 100 m (at constant velocity) to reach the ship, we use x = νt, or
t=\frac{100 m}{0.10 m/s}=1.00 \times 10^3 s =16 \min 40 s.
REFLECT With a 20 minute air supply, she makes it back to the ship safely, with 3 min 20 s to spare.
Practice Problem: If the astronaut has only a 10 minute air supply left, how fast must she throw the wrench so that she makes it back to the spaceship in time? Answer: 17 m/s
