Colliding balls: Conservation of momentum and changes in kinetic energy?

In summary, the conversation discusses the topic of firearm ballistics and the difference between momentum and kinetic energy. It is determined that a faster, lighter bullet causes more damage due to its greater kinetic energy. The conversation then delves into a scenario involving cue balls and the transfer of momentum and kinetic energy during a collision. It is concluded that both conservation of momentum and energy must be considered to determine the final speeds of the balls after the collision.
  • #1
cueballbullet
1
0
I got curious about firearm ballistics and googled something similar to "bullet momentum vs kinetic energy".

IIRC, momentum P = mv (checked); and kE = (mv^2)/2 (also checked).

So I essentially wondered if it's worse to get hit by a bullet with greater kE than by one with lesser kE, presuming that P remains the same (same momentum (also same shape and size); yet different masses and velocities).

Quickly I learned that the faster, lighter bullet causes more damage and has (/because it has) more kE, as the greater amount of kE gets transferred to the bodily tissues.

Cool. Yet this led me to wonder about something else:

Posit that a rolling cue ball, B, of mass M, moving at velocity V, hits another cue ball, b, of mass M/2. If momentum is conserved, then the latter, lighter cue ball, b, will start rolling at velocity 2V... So, same momentum, and different velocities. This means that b has greater kinetic energy than B.

Everything makes sense in my non-physicist mind up until that last sentence. For the life of me I can't guess at all where that extra energy comes from. Same momentum, but twice the speed, because of half the weight. Cool. But again, if the momentum is indeed the same, but the speeds are different, then the kE should also be different, right? How does this work? I may have misunderstood something along the way and perhaps the energy is not greater in b than in B, afterall.
 
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  • #2
cueballbullet said:
Posit that a rolling cue ball, B, of mass M, moving at velocity V, hits another cue ball, b, of mass M/2. If momentum is conserved, then the latter, lighter cue ball, b, will start rolling at velocity 2V... So, same momentum, and different velocities. This means that b has greater kinetic energy than B.
You are assuming that the rolling ball transfers all its momentum to the second ball, then stops dead. That's not how it works. To figure out the speeds of both after the collision, one must apply both conservation of momentum (total momentum of both) and conservation of energy. (If anything, in a real collision, some of the energy will be "lost" to heat and sound.)
cueballbullet said:
Everything makes sense in my non-physicist mind up until that last sentence. For the life of me I can't guess at all where that extra energy comes from.
That's good instinct to sense something's not right. The answer: There is no extra energy!
 
  • #3
Just for fun, here are the final speeds of each. (Assuming a perfectly elastic head-on collision, which is the simplest to analyze.)

Final speed of the first ball: V/3
Final speed of the second ball: 4V/3
 

FAQ: Colliding balls: Conservation of momentum and changes in kinetic energy?

What is conservation of momentum?

Conservation of momentum is a fundamental principle in physics which states that the total momentum of a closed system remains constant over time. This means that in a collision between two objects, the total momentum before the collision is equal to the total momentum after the collision.

How does the conservation of momentum apply to colliding balls?

In the case of colliding balls, the total momentum of the system is equal to the sum of the momentums of the individual balls. This means that the total momentum before the collision is equal to the total momentum after the collision. This principle can be used to predict the velocities of the balls after the collision.

What is kinetic energy and how is it related to momentum?

Kinetic energy is the energy an object possesses due to its motion. It is directly proportional to the mass of the object and the square of its velocity. In the case of colliding balls, the kinetic energy of the system is not conserved as some of it is transferred to other forms of energy, such as sound or heat. However, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

Can the kinetic energy of the colliding balls change during the collision?

Yes, the kinetic energy of the colliding balls can change during the collision. This is because some of the kinetic energy is transferred to other forms of energy, such as sound or heat. However, the total kinetic energy of the system remains constant, in accordance with the principle of conservation of energy.

What factors can affect the outcome of a collision between two balls?

Several factors can affect the outcome of a collision between two balls, including the mass, velocity, and angle of collision of the balls. The elasticity of the balls and any external forces, such as friction, can also affect the outcome of the collision. Additionally, the conservation of momentum and energy must be considered when predicting the outcome of a collision between two balls.

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