The Physics
Hypertextbook
Opus in profectus

# Momentum and Energy

## Problems

### practice

1. 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 attention to the energy and momentum before and after each collision. Does the outcome describe a completely inelastic, partially inelastic, completely elastic, or impossible collision? Provide a brief explanation to accompany each answer.
2. 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. energy-in-collisions-2.pdf
The diagrams below represent generic objects before a collision followed by a set of outcomes to be considered. Comment on the outcomes, paying attention to the energy and momentum before and after each collision. Does the outcome describe a completely inelastic, partially inelastic, 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. The US Army was the first to equip its infantry with semi-automatic or self-loading rifles. From World War II to the Korean War, the standard issue rifle was the M1 Garand, which had a traditional looking soft, sculpted wooden stock. The standard issue rifle from the Vietnam War to the present has been the M16 — a modern looking hard, angular, weapon with lightweight plastic and aluminum components. Both rifles had barrels made of heavy steel. (That's something that will probably never change.) The heavy M1 was loaded with big, heavy .30 '06 (thirty ought six) cartridges. The lighter M16 uses smaller, lighter .223s (two twenty threes). Both shoot bullets at more than twice the speed of sound, the M16's being a little bit faster.
1. Complete the table below.
2. Given the values in the table, what practical advantages does the M16 have over the M1 Garand?
3. The M16 is purported to be more lethal than the M1 Garand. Do your calculations agree with this statement?
M1 Garand M16
years of
service
1936–1957 1964–Present
mass 4.4 kg
(9.6 lb.)
3.8 kg
(7.5 lb.)
barrel
length
610 mm
(24 in.)
510 mm
(20 in.)
caliber 7.62×51 mm
(.30 '06 Springfield)
5.56×45 mm
(.223 Remington)
bullet
mass
9.7 gram
(150 grain)
3.9 gram
(61 grain)
muzzle
velocity
890 m/s
(2900 fps)
950 m/s
(3100 fps)
bullet
energy

bullet
momentum

recoil
velocity

3. 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
4. A 0.020 kg bullet is fired from a gun towards a ballistic pendulum (a large mass supported by cables). The impact speed of the bullet is 400 m/s and the mass of the pendulum is 20.0 kg. The bullet strikes the pendulum and lodges inside.

1. Determine the momentum of the bullet before impact.
2. Determine the kinetic energy of the bullet before impact.
3. Determine the momentum of the pendulum after impact.
4. Determine the speed of the pendulum after impact.
5. Determine the kinetic energy of the pendulum after impact.
6. How high does the pendulum rise?

### 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.

### 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.
2. A billiard ball of mass 0.10 kg moving to the right at 3.0 m/s collides head on with an identical stationary billiard ball. The magnitude of the collision force (in newtons) is given by the following functions of time (in seconds) for 0 ≤ t ≤ 0.01 s…
 F(t) = 40(1 − 200|t − 0.005|)F(t) = 60(1 − 200|t − 0.005|)F(t) = 1.2 × 106t(0.01 − t)F(t) = 1.8 × 106t(0.01 − t)
After colliding, the two balls separate. For each function, determine…
1. the magnitude of the total impulse of the collision
2. the change in momentum of the moving ball (magnitude and direction)
3. the change in momentum of the stationary ball (magnitude and direction)
4. the final velocity of the moving ball (magnitude and direction)
5. the final velocity of the stationary ball (magnitude and direction)
6. the type of collision: elastic, inelastic, or superelastic (justify your answer)

### statistical

1. The table at the bottom of this problem, which is also available as a text file, shows the characteristics of some handguns.
1. For each weapon, determine…
1. the recoil velocity of the gun.
2. the kinetic energy of the bullet.
3. the kinetic energy of the gun.
4. the percent of the total kinetic energy carried by the bullet.
2. According to the calculations above, which weapon(s) is…
2. the easiest to handle? the hardest to handle?
3. the most efficient at projecting bullets? the most wasteful?
3. The bullet strikes a stationary 60 kg person and lodges inside them. For each bullet, determine…
1. the velocity of the victim after imapact.
2. the kinetic energy of the victim after impact.
3. the percent of the bullet's kinetic energy lost during impact.
4. According to the calculations above, which bullet(s) is…
1. the most efficient at causing harm? the least efficient?
Characteristics of some handguns
name type gun
mass (g)
cartridge bullet
mass (g)
bullet
velocity (m/s)
1. Beretta
Pico
semi­automatic pistol 330 .380 ACP 6 300
2. Browning
BDA
semi­automatic pistol 920 9 mm IMI 8 360
3. Colt
Anaconda
revolver 1,300 .44 Magnum 16 360
4. Colt
Detective Special
revolver 595 .38 Special 7 300
5. Colt
M1892
revolver 940 .38 Long Colt 8 235
6. COP
.357
derringer 800 .357 Magnum 8 440
7. FN Herstal
Five-seveN
semi­automatic pistol 652 5.7×28 mm 1.8 620
8. IMI
Desert Eagle
semi­automatic pistol 1,560 .50 Action Express 21 427