Momentum & Energy

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Discussion

Types of collisions.

Energy in Collisions
type total
kinetic energy		 comments
perfectly inelastic decreases to a minimum objects stick together
inelastic decreases by any amount all collisions between macroscopic bodies
partially elastic or nearly elastic "nearly
conserved"		 billiard balls, bowling balls, steel bearings and other objects made from resilient materials
elastic absolutely conserved collisions between atoms, molecules, subatomic particles and other similar microscopic bodies
superelastic increases contrived collisions between objects that release potential energy on contact, fictional superelastic materials like flubber

Anything else?

Summary

Problems

practice

  1. Work along with this example using the worksheet energy-in-collisions-1.pdf.
    The diagrams below represent generic objects before a collision followed by a set of outcomes to be considered. Comment on the outcomes, paying strict attention to the energy and momentum of each system before and after the collision. Does the outcome describe a completely inelastic, partially inelastic, partially elastic, completely elastic, or impossible collision? Provide a brief explanation to accompany each answer. (Note: in order to conserve space, the masses and velocities were not drawn to scale.) Solutions …
    1. Here we are given the initial conditions of the two colliding objects.
       
       
      ∑ p =    m1v1  +    m2v2  =    (8 kg)(+6 m/s)  +    (4 kg)(−9 m/s)  = +12 kg·m/s
             
      ∑ K =  1  m1v12  +  1  m2v22  =  1  (8 kg)(6 m/s)2  +  1  (4 kg)(9 m/s)2  = 306 J
      2 2 2 2
       
      Momentum is a vector quantity, so the total momentum is found by a vector sum. Since the momentums of the two objects are in opposite directions one of them is going to be negative. Since positive answers are preferred over negative ones, let's choose right as the positive direction. This gives us a total momentum of +12 kg·m/s. Energy being a scalar is much easier to handle, especially here since the only energy that matters is kinetic (which is always positive). The total mechanical energy of this system in 306 J.
    2. The first outcome we'll be examining has the two objects sticking together and moving off to the right.
       
       
      ∑ p' =    (m1 + m2)v'  =    (8 + 4 kg)(+1 m/s)  = +12 kg·m/s
         
      ∑ K' =  1  (m1 + m2)v'2  =  1  (8 + 4 kg)(1 m/s)2  = 6 J
      2 2
       
      This is a sensible outcome since the initial total momentum is positive (to the right). Calculations show that the final total momentum is still +12 kg·m/s, but the final total energy has dropped significantly to a relatively low value of 6 J. Momentum was conserved, but mechanical energy was lost. This a classic example of an inelastic collision. (Some would even call this a completely inelastic collision.) The lost energy has likely gone into plastic deformation of the two objects (given the distorted edges shown in the diagram).
    3. Here the two objects have separated after collision and are moving in opposite directions. Each is moving more slowly than it was before the collision. This hints at a loss of mechanical energy.
       
       
      ∑ p' =    m1v1'  +    m2v2'  =    (8 kg)(−1 m/s)  +    (4 kg)(+5 m/s)  = +12 kg·m/s
             
      ∑ K' =  1  m1v1'2  +  1  m2v2'2  =  1  (8 kg)(1 m/s)2  +  1  (4 kg)(+5 m/s)2  = 54 J
      2 2 2 2
       
      Momentum was conserved as it should be, but mechanical energy was lost making this an inelastic collision. Since more energy was retained than in the previous outcome, some would call this a partially inelastic collision. Lost energy is not a big deal and does not violate the conservation of energy. The energy wasn't destroyed in this outcome. It just turned into a form that isn't easy to see with our eyes -- internal energy. When kinetic energy transforms into internal energy, either the temperature of the system increases or it experience a phase change (melting, for example). Internal energy will be dealt with in more detail later in this book.
    4. This outcome is similar to the previous one only now the objects are moving a bit more quickly. Still, their speeds after the collision are slower than before.
       
       
      ∑ p' =    m1v1'  +    m2v2'  =    (8 kg)(−3 m/s)  +    (4 kg)(+9 m/s)  = +12 kg·m/s
             
      ∑ K' =  1  m1v1'2  +  1  m2v2'2  =  1  (8 kg)(3 m/s)2  +  1  (4 kg)(9 m/s)2  = 198 J
      2 2 2 2
       
      Momentum was conserved and energy was lost, but to a lesser extent than in the previous outcome. Of all the outcomes so far, this inelastic collision is the least inelastic. Whether one would call it partially inelastic or partially elastic doesn't really matter.
    5. Here we see an elastic collision. (Some would even call this a perfectly elastic collision.)
       
       
      ∑ p' =    m1v1'  +    m2v2'  =    (8 kg)(−4 m/s)  +    (4 kg)(+11 m/s)  = +12 kg·m/s
             
      ∑ K' =  1  m1v1'2  +  1  m2v2'2  =  1  (8 kg)(4 m/s)2  +  1  (4 kg)(11 m/s)2  = 306 J
      2 2 2 2
       
      Momentum and total mechanical energy of the system were both conserved. The total kinetic energy of the two objects after the collision is the same as it was before. In the macroscopic world such an outcome would never happen. The results above show the limit of what could happen. That is, macroscopic objects will always have less total mechanical energy after a collision than before. Never equal to or greater than.
    6. This outcome is difficult to explain.
       
       
      ∑ p' =    m1v1'  +    m2v2'  =    (8 kg)(−6 m/s)  +    (4 kg)(+15 m/s)  = +12 kg·m/s
             
      ∑ K' =  1  m1v1'2  +  1  m2v2'2  =  1  (8 kg)(6 m/s)2  +  1  (4 kg)(15 m/s)2  = 594 J
      2 2 2 2
       
      Momentum was conserved, but mechanical energy increased. How could this happen? Where did the extra energy come from? Since this is a problem about generic objects, we are free to contrive all sorts of explanations. Perhaps there was a compressed spring on one of the objects, or a chemical explosive, or the two objects were small mammals that kicked off of each other after they collided. I think "contrive" was an appropriate word choice on my part to describe what we're doing here. This outcome seems improbable as it violates the conservation of mechanical energy in a way different from the previous outcomes.
    7. This outcome is also perplexing.
       
       
      ∑ p' =    m1v1'  +    m2v2'  =    (8 kg)(−8 m/s)  +    (4 kg)(+5 m/s)  = −44 kg·m/s
             
      ∑ K' =  1  m1v1'2  +  1  m2v2'2  =  1  (8 kg)(8 m/s)2  +  1  (4 kg)(5 m/s)2  = 306 J
      2 2 2 2
       
      Mechanical energy was conserved in this outcome (which is rare, but not impossible), but momentum was not. This would require an impulse applied from outside of the system. Since the description of this problem makes no mention of the existence or even the possibility of a third object, we would have to conclude that this outcome is impossible as it violates the conservation of linear momentum.
  2. Follow along with this activity using the worksheet bowling-balls.pdf.
    1. A 5 kg bowling ball moving at 8 m/s approaches a row of stationary balls lined up end to end in a ball return. Comment on the likelihood of the following outcomes.
      1. The incoming ball stops and one 5 kg ball leaves the row of stationary balls at a speed of 8 m/s.
      2. The incoming ball stops and two 5 kg balls leave the row of stationary balls at a speed of 4 m/s.
    2. Two 5 kg bowling balls moving at 8 m/s approach a row of stationary balls lined up end to end in a ball return. Comment on the likelihood of the following outcomes.
      1. The incoming balls stop and two 5 kg balls leave the row of stationary balls at a speed of 8 m/s.
      2. The incoming balls stop and one 5 kg ball leaves the row of stationary balls at a speed of 16 m/s.
    Compute the total linear momentum and mechanical energy of the bowling balls before and after each collision has occurred. Compare the values of these quantities to answer this question.
    1. In the case of one bowling ball the initial momentum and energy are …
       
       
      1. In the first hypothetical outcome shown below, both momentum and energy are completely conserved. Such a collision is said to be perfectly elastic. While totally fine from a theoretical perspective, such an outcome is practically impossible. In the macroscopic real world, momentum and energy are always dissipated. (In the microscopic world of atoms and molecules collisions are always elastic, but that is another story.)
         
         
        This outcome, while highly improbable, is not theoretically impossible.
      2. In this hypothetical outcome, momentum is conserved but mechanical energy is lost. Such a collision is said to be inelastic. While totally fine from a theoretical standpoint I think, from the commonsense standpoint of everyday observations of bowling balls in a ball return, that such an outcome is highly unlikely.
         
         
        Of the two outcomes presented neither will occur. The real world doesn't work out according to hypothetical ideals. The real questions should be, "Is the real outcome of this collision more like the first hypothetical outcome or the second?" Experimental observation confirms that the collisions between bowling balls are more like elastic collisions than inelastic collisions. If one bowling ball comes into a row of stationary balls, the most likely outcome is that one bowling ball will leave the other end.
    2. In the case of two bowling balls the initial momentum and energy are …
       
       
      1. This is a trivial solution to the problem. Obviously both momentum and energy are conserved. This is another example of a perfectly elastic collision.
         
         
        This outcome is possible, but not probable.
      2. This last possible outcome makes no sense. The momentum after collision is the same as before, but the mechanical energy has somehow increased. Miraculously, it doubled.
         
         
        While energy can neither be created nor destroyed, it certainly can become "lost". This is why I have no problem with the second outcome of the first collision. However, this outcome is surely impossible. The kinetic energy of a system cannot increase without work being done by some outside agent. For a row of bowling balls sitting in a ball return I cannot see anyway for positive work to be done. Thus if two bowling balls approach a row of stationary balls the most likely outcome is that two will emerge from the other end.
    In general, the collision between bowling balls is more like an elastic collision than an inelastic one. If one ball approaches a row of stationary balls, one ball will leave from the other side. If two balls approach, then two will leave. If three balls approach, three will leave. And so on. This rule is the basis for a popular desktop ornament (and when I say "desktop" I mean the top of a real desk, not the pattern on a computer monitor that one sees when no applications are running or documents are open.). Often called "Newton's cradle" or (more hilariously) "Newton's balls" and identified by a whole host of trade names, it is standard issue for the executive desktop in movies and television. In fact, I once heard it called "the executive intelligence tester". The dialog below illustrates this application.
         
    HR Manager:   Watch. One goes in. One comes out. Two go in. Two come out. Three go in. Three come out. Four go in. What happens next?
    Potential Executive:   Uh … Four come out?
    HR Manager:   Good. Now five go in.
    Potential Executive:   Oo, oo, I know. Five come out!
    HR Manager:   Congratulations, you're our new vice president.


  3. gun & bullet, bullet & target
    • Answer it.
  4. Determine the final velocities of two objects after a one dimensional (head on) elastic collision in terms of the objects' masses and initial velocities. Solution …
    • Start with statements of conservation of momentum and energy.
      Obviously, we can remove the common fraction ½ from each term in the conservation of energy statement. Less ovious, however, it the fact that this equation can be rearranged into the difference of two squares. Expanding this is the key to an easy solution.
      Rearrange the conservation of momentum statement into a similar form.
      Put the two new statements together.
      And then what?

numerical

  1.   top:  .308 caliber, M1 & M14
      bottom:  .223 caliber, M16
     
    The US Army was the first to equip its infantry with semi-automatic (or self-loading) rifles. In World War II, the standard issue rifle was the M1 Garand (not to be confused with the smaller, less popular M1 Carbine). Lessons learned from combat experience lead to several modifications and the M1 gradually evolved into the M14 -- a weapon that was mass produced for the US military and its NATO allies until the beginning of the Vietnam War. The standard issue rifle that followed was a very different weapon -- the M16. The soft, sculpted look traditionally associated with rifles and shotguns was tossed out in favor of a hard, angular, modern style. All the wood and some of steel found on the M1 and M14 were replaced with lightweight plastic and aluminum in the M16. (The barrel is still made of steel for obvious reasons.) The M16 also uses small caliber bullets that are surprisingly more deadly than the big bullets fired by the M1 and M14.
    1. Complete the following table.
    2. Given the values you calculated, answer the following questions …
      1. What practical advantage does the M16 have over its predecessors?
      2. Why should it be "surprising" that the M16 is more deadly than its predecessors?

      M1 Garand M14 M16
    years of
    service    		 1936-1957 1957-1964 1964-????
    mass 4.4 kg
    (9.6 lb.)    		 4.5 kg
    (9.9 lb.)    		 3.8 kg
    (7.5 lb.)
    barrel
    length    		 610 mm
    (24 in.)    		 560 mm
    (22 in.)    		 510 mm
    (20 in.)
    caliber 7.62 x 51 mm
    (.30 '06)    		 7.62 x 51 mm
    (.308 Winchester)    		 5.56 x 45 mm
    (.223 Remington)
    bullet
    mass    		 9.7 g
    (150 grain)    		 9.7 g
    (150 grain)    		 3.6 g
    (55 grain)
    muzzle
    velocity    		 890 m/s
    (2900 f.p.s.)    		 860 m/s
    (2800 f.p.s.)    		 950 m/s
    (3100 f.p.s.)
    bullet
    momentum    		      
    bullet
    energy    		      
    recoil
    velocity    		      

  2. Scientists at Brookhaven National Laboratory in New York in conjunction with Brooklyn Union Gas (now a division of Keyspan Energy) are developing a compressed helium projectile launcher called the RAPTOR (short for "rapid cutter of concrete"). The original technology behind the gas gun began in the 1980s as part of an anti-missile research program. Now instead of shooting down missiles in midair, the RAPTOR will be used to shoot tiny metal projectiles at the ground to cut concrete like a jackhammer. The device works by rapidly compressing helium gas from its storage tank pressure of 2 atmospheres to an unbelievable 1000 atmospheres in a fraction of a second. The resulting shock wave blasts the 1.8 g projectiles (about the same mass as a .22 caliber bullet) out the barrel of the gun at roughly 1600 m/s (more than twice the muzzle velocity of a high-powered rifle). The main benefit of this technology is that it is much quieter than conventional concrete cutters -- 85 dB for the RAPTOR compared to 125 dB for a jackhammer. The last reported prototype (RAPTOR III) was 2.0 m long, weighed 120 kg, and was able to split a 10 cm thick slab in seven shots. Determine …
    1. the work done by the compressed helium on a projectile,
    2. the average force of the compressed helium on a projectile,
    3. the impulse delivered to a projectile,
    4. the time a projectile spends in the barrel,
    5. the recoil speed of the gun,
    6. the height to which the gun would jump, and
    7. the minimum energy needed to split the concrete slab.

algebraic

  1. Show that when a moving object collides elastically with a stationary object, the two velocities after collision will be perpendicular to one another.

calculus

  1. Prove that a "perfectly inelastic" collision occurs when two objects stick together. That is, show that two colliding objects obeying the law of conservation of momentum have a minimum total kinetic energy when they move with the same velocity.

worksheets

  1. energy-in-collisions-2.pdf
    The diagrams on the accompanying pdf worksheet represent generic objects before a collision followed by a set of outcomes to be considered. Comment on the outcomes, paying strict attention to the energy and momentum of each system before and after the collision. Does the outcome describe a completely inelastic, partially inelastic, partially elastic, completely elastic, or impossible collision? Provide a brief explanation to accompany each answer. (Note: in order to conserve space, the masses and velocities were not drawn to scale.)

Resources

prev | up | next

Another quality webpage by

Glenn Elert		 eglobe logo home | contact

bent | chaos | eworld | facts | physics