Momentum & Energy

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.)
  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.
  3. This is a gruesome question about the effectiveness of handguns when used for their intended purpose. A 1.10 kg pistol fires a 10.1 g bullet with a muzzle velocity of 375 m/s. Determine …
    1. the kinetic energy of the bullet
    2. the recoil velocity of the pistol
    3. the kinetic energy of the pistol
    4. the fraction of the total energy delivered to the bullet
    The bullet strikes a 75 kg stationary man and does not come out the other side. Determine …
    1. the velocity of the man and bullet together
    2. the kinetic energy of the man and bullet together
    3. the fraction of the total kinetic energy lost in the collision
  4. Two objects (m1 and m2) traveling in opposite directions (+v1 and v2) collide head on and stick together. Derive an expression for …
    1. the final velocity of the two objects stuck together (easy)
    2. the kinetic energy lost as a result of the collision (hard)

numerical

  1. Rifle Bullets
    left: .308 caliber, M1 & M14
    right: .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-Present
    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)
    3.9 g
    (61 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
    7. the minimum energy needed to split the concrete slab

algebraic

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

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.)

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.