Conservation of Momentum
Discussion
start
Start with Newton's third law of motion, toss in the impulsemomentum theorem, and see what happens.
+F_{1} =  −F_{2} 
+F_{1}Δt =  −F_{2}Δt 
+m_{1}Δv_{1} =  −m_{2}Δv_{2} 
+Δp_{1} =  −Δp_{2} 
+(p_{1}′ − p_{1}) =  −(p_{2}′ − p_{2}) 
p_{1}′ + p_{2}′ =  p_{1} + p_{2} 
∑p′ =  ∑p 
Or flip it around.
∑p = ∑p′
The total momentum before an interaction is the same as the total momentum after the interaction. Momentum is conserved. The total momentum of a closed system is constant.
1668: John Wallis suggests the law of conservation of momentum
recoil
Action is equal and opposite reaction.
collisions
Write something.
changing mass problems
Mass in and/or mass out.
∑F =  dp  = m  dv  + v  dm 
dt  dt  dt 
There are two typical problems given to first year physics students that use this variation on Newton's second law of motion: conveyor belts and rockets.
Rockets first. The father of the academic study of rocketry and space travel is Konstantin Tsiolkovsky (1857–1935) Russia. Tsiolkovskiy was a Nineteenth Century visionary who anticipated many features of Twentieth Century space exploration.
Start with Newton's second law of motion. The net external force on the rocket is zero. Nothing is pushing the spacecraft but itself.
∑F =  dp  = m  dv  + v  dm  = 0 
dt  dt  dt 
Thus, momentum is conserved. We begin with a fancier looking version of the actionreaction or recoil problem.
m  dv  = − v  dm 
dt  dt 
Even though the denominators on both sides are infinitesimals, they still cancel out. We'll also drop the vector notation since this is a onedimensional problem at it's heart.
+ m dv = − v dm
Apply the technique of separation of variables.
+  ⌠ ⌡ 
dv = −v_{exhaust}  ⌠ ⌡ 
dm 
m 
Integrate over the appropriate limits.
+ (v_{final} − v_{initial}) = −v_{exhaust} (log m_{final} − log m_{initial})
Clean it up a bit.
Δv_{rocket} = v_{exhaust} log  ⎛ ⎝ 
m_{initial}  ⎞ ⎠ 
m_{final} 
We could quit here, but it's tradition to write the variables in a certain way.
Δv_{rocket} = v_{exhaust} log  ⎛ ⎝ 
m_{empty rocket} + m_{fuel}  ⎞ ⎠ 

m_{empty rocket} 
Clean it up a bit more.
Δv_{rocket} = v_{exhaust} log  ⎛ ⎝ 
1 +  m_{fuel}  ⎞ ⎠ 

m_{empty rocket} 
This is Tsiolkovskiy's rocket equation, which was first published in 1903 — the year the Wright brothers flew the first airplane, 23 years before Robert Goddard built the first liquid fuel rocket, 54 years before the Soviet Union placed the first artificial satellite in orbit around the Earth, and 66 years before the United States sent humans to play golf on the moon. The rocket equation appeared in an essay with an odd sounding title in the 1968 English translation — "Investigation of world spaces by reactive vehicles" (in Russian, "Исследование мировых пространств реактивными приборами"). "World spaces" would now be called "outer space" and "reactive vehicles" is an overly literal translation of a phrase that really means something like "jet propulsion". A better English translation of the title would probably be "Exploration of outer space by jet propulsion".
An interesting aside. Tsiolkovskiy saw rocket trips to outer space in his mind's eye, but missed seeing radio as the medium for communicating over large distances. (The rocket equation was written three years before the first AM radio broadcast.) When Tsiolkovskiy pondered how cosmonauts would stay in touch with earth he thought they might use special rockets to send messages back and forth. In our current age, when people spend more time talking to disembodied voices on mobile phones than talking face to face, it's hard to imagine communicating over astronomical distances using the equivalent of rocket mail trucks or space carrier pigeons.
And now for conveyor belts…. Eh… it's kind of a let down after rockets. I think I'll end here and write a conveyor belt question for the practice problems section later.