an astronaut on an eva has wandered dangerously far away from the shuttle. she has also exhausted all the fuel in her jet pack. how can she get back to the shuttle?

The conservation of the momentum allows to find the result of how the astronaut can return to the spacecraft is:

The momentum is defined as the product of the mass and the velocity of the body, for isolated systems the momentum  is conserved. If we define the system as consisting of the astronaut and the evo propellant, this system is isolated and the internal forces become zero. Let's find the moment in two moments.

Initial instant. Astronaut and thrust together.

        p₀ = 0

Final moment. The astronaut now the thruster in the opposite direction of the ship.

        = m v + M v '

where m is propellant mass and M the astronaut mass.

As the moment is preserved.

       0 = m v + M v ’

      v ’=  

We can see that the astronaut's speed is in the opposite direction to the propeller, that is, in the direction of the ship.

The magnitude of the velocity is given by the relationship between the masses.

In conclusion, using the conservation of the momentun we can find the result of how the astronaut can return to the ship is:

Learn more here:  

e/c, where e is its total energy

could you add the photo?

explanation: