An astronaut with a mass of #95 kg# is floating in space. If the astronaut throws an object with a mass of #4 kg# at a speed of #1/8 m/s#, how much will his speed change by?
1 Answer
This is looking at conservation of momentum.
Since the astronaut is floating in space, there is the assumption of no gravity and no air resistance, so as you throw an object forward, you should move backwards.
According to Newton's Third Law, when you push the object, it pushes you back. Here, the astronaut and object move in opposite directions. Hence, if we say the object is thrown to the right, we have conservation of momentum like so:
#Deltap_o - Deltap_a = 0#
#vecp_(o,i) - vecp_(o,f) = vecp_(a,i) - vecp_(a,f)#
#vecp_(o,i) - vecp_(a,i) = vecp_(o,f) - vecp_(a,f)#
#cancel(m_(o,i)vecv_(o,i)) - cancel(m_(a,i)vecv_(a,i)) = m_(o,f)vecv_(o,f) - m_(a,f)vecv_(a,f)# where
#o# is object,#a# is astronaut, and#i# and#f# are initial and final, respectively. Both the astronaut and the object start at rest, so initial momenta are#0# , and we've defined rightwards motion as positive.
Thus:
#m_(a,f)vecv_(a,f) = m_(o,f)vecv_(o,f)#
#v_(a,f) = |(m_(o,f)v_(o,f))/(m_(a,f))|#
#color(blue)(v_(a,f)) = (("4 kg")("0.125 m/s"))/(("95 kg"))#
#color(blue)"= 0.0053 m/s"#
in the opposite direction.
Assuming that the astronaut started at rest, the change in speed is just his new speed.
