Question

an astronaut having mass 320kg with equipment included is attempting an untethered space walk. the astronaut is initially at rest with respect to the nearby spacecraft, but discharges 30.0g of nitrogen gas at 900.0 m/s from a portable unit. how fast is the astronaut moving after doing that?

Answer 1

Let's think about the system that includes the spacecraft, the astronaut,
and all the equipment tied to him.  In this system, nothing is moving relative
to anything else, so the total linear momentum is zero.

Now, suddenly, a blast of gas leaves the astronaut's little SCUBA tank, and
hisses away from him and the spacecraft, in that direction  ===> .

The momentum of the cloud of nitrogen is

                         (mass)  x  (speed)

                     =  (0.03 kg)  x  (900 m/s)

                     =     27  kg-m/s         in that direction  ===>  . 

Now is the perfect time to recall that momentum is conserved.
It can't be suddenly created or destroyed, it can't appear out of
nowhere, it can't disappear into nowhere, and the total amount
of momentum in a system is constant.

In order to keep the total momentum of this system constant, the
astronaut himself gets  27 kg-m/s of momentum in this direction  <=== .

The mass of him and his equipment is 320 kg.
So ...

                27 kg-m/s  <==  =  (320 kg) x (speed <==)

                Speed  =  27 kg-m/s  /  320 kg

                             =   0.084375 m/s   in this direction  <===