Conservation of Momentum

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

start

Start with Newton's third law of motion, toss in the impulse-momentum theorem, and see what happens.

  +F1 =  −F2  
  +F1Δt =  −F2Δt  
  +m1Δv1 =  −m2Δv2
  +Δp1 =  −Δp2  
  +(p1 − p10) =  −(p2 − p20)  
  p1 + p2 =  p10 + p20  
  ∑p =  ∑p0  

Momentum is conserved. The total momentum of a closed system is constant.

recoil

The symbol p may be used for momentum because it is the mirror image of the letter q. Some people use p to represent the momentum of one object and q to represent the momentum of the other object in a recoil situation. A clever way to write Newton's third law of motion is like this …

p = q

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 Eduardovich Tsiolkovskiy a.k.a. Константин Эдуардович Циолковский a.k.a. Konstanty Ciołkowski (1857-1935) Russia. Tsiolkovskiy was an exceptional visionary who anticipated many features of the late Twentieth Century space programs in the Nineteenth Century.

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 action-reaction 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 one-dimensional problem at it's heart.

+ m dv = − v dm

Apply the technique of separation of variables.

+  ⌠
⌡		 dv = −vexhaust ⌠
⌡		 dm
m

Integrate over the appropriate limits.

+ (vfinal − vinitial) = −vexhaust (log mfinal − log minitial)

Clean it up a bit.

Δvrocket = vexhaust log ⎛
⎜
⎝		 minitial ⎞
⎟
⎠
mfinal

We could quit here, but it's tradition to write the variables in a certain way.

Δvrocket = vexhaust log ⎛
⎜
⎝		 mempty rocket + mfuel ⎞
⎟
⎠
mempty rocket

Clean it up a bit more.

Δvrocket = vexhaust log ⎛
⎜
⎝		 1 +  mfuel ⎞
⎟
⎠
mempty 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 astronauts or 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 a cell phone 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.

Summary


  I II
1st law inertia
m		 momentum
p = mv
2nd law force law
F = m a          F = m dv / dt		 impulse-momentum theorem
J = Δp          ∫ Fdt = m ∫ dv
3rd law action-reaction
F1 = -F2		 conservation of momentum
∑ p = ∑p0

Problems

practice

  1. In the action-adventure movie Eraser, (1996) Arnold Schwarzenegger plays a US Marshall working for the Witness Relocation Program. He obtains a high-tech weapon that launches projectiles at "nearly the speed of light." Arnold wields the weapon like a shotgun, firing it several times from an unbraced, standing position. Bad guys are then seen flying through the air (for several meters) after being hit. Explain the physical impossibility of operating such a weapon in the manner described above.
    • Answer it.
  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write a conveyor belt problem and make it interesting.
    • Answer it.

numerical

  1. A 3.0 kg fish is swimming at 1.5 m/s to the right. It swallows a 0.25 kg fish swimming to the left at 4.0 m/s. What is the velocity of the larger fish immediately after lunch?

calculus

  1. A rocket or some other kind of space travel problem.
  2. Another conveyor belt problem maybe.

statistical

  1. student-data.txt
    The picture below shows an experiment performed by a group of students. Two lab carts are placed end to end in the center of an aluminum track. The cart on the left has a trigger that will release a spring-loaded piston when pressed. This pushes the two carts in opposite directions across the track. The plastic cards on top of each cart then passed through the photogates on opposite ends and a computer computed the speeds of the carts.

    The students placed various combinations of iron weights on the carts and repeated the experiment several times. The mass (in kg) of each cart plus the weights and the recoil speeds (in cm/s) were recorded in the accompanying tab delimited text file.
    1. Compute the recoil momentums of each cart.
    2. Construct a graph with the momentum of the first cart on the horizontal axis and the momentum of the second cart on the vertical axis.
    3. Add a best fit straight line to this graph and determine its slope and y-intercept. (Include the uncertainty, coefficient of correlation or determination, and the root mean square error if you have the ability to do so.)
    4. Does this analysis show that momentum was conserved when the lab carts recoiled?

investigative

  1. Determine "delta vee" for a typical Space Shuttle from SRB separation to MECO. Begin by finding the following information …
    1. mass of the orbiter
    2. mass of the external fuel tank …
      1. full
      2. empty
    3. time after liftoff of …
      1. solid rocket booster (SRB) separation
      2. main engine cutoff (MECO)
    4. exhaust velocity of the main engines
    5. type of fuel used in the main engines
    6. the Space Shuttle's "destination" in space
  2. Determine "delta vee" for either the Deep Space 1 or Dawn space probes after leaving the earth's gravitational field. Begin by finding the following information …
    1. mass of the probe when …
      1. full of propellant
      2. empty
    2. exhaust velocity
    3. type of propellant used
    4. engine burn time
    5. where the probe was sent

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