An astronaut wearing a space suit floats in the middle of a room with no air (i. e. vacuum) and is 20.0 m from any wall, the floor and the ceiling. She needs to get to the door so she can go with her NASA friends to eat a hamburger. She has only a book and a pencil in her possession. She cannot call for help since something is interfering with her signal. Under these conditions, how can she get to the door

To get to the door, the astronaut has to throw the book and pencil away from her in the direction opposite to that of the door.

Explanation:

Since no net force acts on her and she floats in the center of the room, if she throws the book and pencil away from her, since both her, the book and the pencil have an initial momentum p = 0, and since momentum is conserved, then after throwing the book and pencil away from her, the momentum of both her, the book and pencil is still zero.

The astronaut will thus develop a momentum which is opposite to that of the book and pencil. This is shown below.

Since p = p' where p = initial momentum of astronaut, book and pencil = 0 and p' = final momentum of astronaut, book and pencil = mv + m'v' where m = mass of book and pencil, v = velocity of book and pencil, m' = mass of astronaut and v' = velocity of astronaut.

So, p = p'

0 = mv + m'v'

mv = -m'v' where mv = momentum of book and pencil and m'v' = momentum of astronaut

v' = -mv/m

We see that the astronaut develops a momentum opposite in direction but equal in magnitude to that of the book and pencil and will thus develop a velocity opposite to that of the book and pencil.

Thus, to get to the door, the astronaut has to throw the book and pencil away from her in the direction opposite to that of the door.