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

• B
• cueballbullet
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.

#### cueballbullet

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.

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!

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

## 1. What is the law of conservation of momentum?

The law of conservation of momentum states that in a closed system, the total momentum before a collision is equal to the total momentum after the collision. This means that the total amount of momentum in a system remains constant, even if there is a collision or interaction between objects.

## 2. How is the law of conservation of momentum applied in the context of colliding balls?

In the context of colliding balls, the law of conservation of momentum means that the total momentum of the balls before the collision is equal to the total momentum of the balls after the collision. This means that the total mass and velocity of the balls will remain constant.

## 3. What is kinetic energy and how does it change during a collision?

Kinetic energy is the energy an object possesses due to its motion. During a collision, some of the kinetic energy of the colliding balls is converted into other forms of energy, such as heat or sound. However, the total amount of kinetic energy in the system is conserved, meaning that the sum of the kinetic energies before and after the collision will be equal.

## 4. Can the law of conservation of momentum and the law of conservation of energy be violated?

No, these laws are fundamental principles in physics and have been extensively tested and proven through experiments. They are considered to be universal laws that apply to all physical systems, and have not been found to be violated in any observed phenomena.

## 5. How do elastic and inelastic collisions differ in terms of conservation of momentum and kinetic energy?

In an elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum and total kinetic energy before the collision will be equal to the total momentum and total kinetic energy after the collision. In an inelastic collision, some of the kinetic energy is converted into other forms of energy, so the total kinetic energy after the collision will be less than the total kinetic energy before the collision. However, the law of conservation of momentum still holds true in both elastic and inelastic collisions.