Why conservation of energy here vs momentum?

In summary: But that's why we need to use both conservation of momentum and conservation of energy to solve for the final velocities of both balls.
  • #1
oddjobmj
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0

Homework Statement



A billiard ball moving at 2.58 m/s collides elastically with an identical ball initially at rest. After the collision the speed of one ball is 1.36 m/s. What is the speed of the other?

Homework Equations



K1+K2=K1'+K2'

or

P1+P2=P1'+P2'

The Attempt at a Solution



I get the correct answer using conservation of energy and the wrong answer using conservation of momentum. My understanding is that conservation of momentum problems work for both elastic and inelastic problems whereas conservation of energy work for elastic problems only (unless energy lost is known).

In this case the masses cancel out so both methods (momentum/energy) actually look the same when solved symbolically other than the energy equation yields squared velocities and the momentum equation does not.

How do I know to use conservation of energy here instead of conservation of momentum? Shouldn't momentum also be conserved?

Thank you!
 
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  • #2
Out of curiosity, what are the two answers that you are getting? Being that they are identical, and I am assuming you mean that the collision is completely elastic, conservation of momentum would give you a speed of 1.22, correct?
 
  • #3
Yes, conservation of momentum yields 1.22 m/s. Conservation of energy yields 2.19 m/s. The correct answer is 2.19 m/s.

I guess it does not say 'perfectly elastically' but just elastically.
 
  • #4
2.19 m/s would assume an increase in overall kinetic energy, which seems very unlikely without the mass of the first ball being larger than that of the second.

Edit: Sorry, I fat fingered this on my calculator. 2.19 by kinetic energy balance is correct, without an increase in overall energy.
 
Last edited:
  • #5
Nah, that is the result when using conservation of energy. Because (1/2)'s and masses cancel out:

v1i2=v1f2+v2f2

v1i2-v1f2=v2f2

v2f=[itex]\sqrt{v_{1i}^2-v_{1f}^2}[/itex]
 
  • #6
oddjobmj said:
How do I know to use conservation of energy here instead of conservation of momentum? Shouldn't momentum also be conserved?
Momentum is conserved! But remember that it is a vector. How can you calculate the final momentum of each ball if you don't know the angle between the trajectories of the two balls after collision?
 
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  • #7
Ah, find the magnitude of the resulting vector using the Pythagorean theorem which ends up looking the same as the energy equation.

Strange though that there is no indication or reason to believe they wouldn't both be traveling in the initial direction.

edit: maybe there is? I guess if they were identical balls and the resulting velocity was in the direction of the initial velocity the first ball would stop and the second ball would travel at the initial velocity, right?
 
  • #8
Think of it as one ball hitting the other ball off center. If you have the first ball moving only in the x direction, and the ball nicks the bottom of the second ball, then both balls will end up with y components.
 
  • #9
If a problem involves billiard balls, you should never assume that you can consider the system to be one-dimensional unless specifically told otherwise.
 
  • #10
oddjobmj said:
Yes, conservation of momentum yields 1.22 m/s.
No, it doesn't. You don't have enough information to solve for the velocity of the other ball using conservation of momentum.

Conservation of energy yields 2.19 m/s. The correct answer is 2.19 m/s.
That is correct.

I guess it does not say 'perfectly elastically' but just elastically.
"Perfectly elastic" and "elastic" are synonyms. The term means the collision conserves kinetic energy.
 
  • #11
oddjobmj said:
Strange though that there is no indication or reason to believe they wouldn't both be traveling in the initial direction.

edit: maybe there is? I guess if they were identical balls and the resulting velocity was in the direction of the initial velocity the first ball would stop and the second ball would travel at the initial velocity, right?

You obviously never seen a game of billiards!
 
  • #12
I am happy to proceed with the assumption that they are not moving in the same direction if it is not explicitly stated so. However, I guess we do actually have enough information to say that they definitely are not moving in the same direction simply because they are both moving. If the collision were perfectly elastic and it was a one dimensional problem the first ball would stop and the second ball would proceed at the initial ball's velocity.

Thank you for your help!

edit: Hah, PeroK, thanks.
 
  • #13
oddjobmj said:
However, I guess we do actually have enough information to say that they definitely are not moving in the same direction simply because they are both moving. If the collision were perfectly elastic and it was a one dimensional problem the first ball would stop and the second ball would proceed at the initial ball's velocity.
Exactly.
 

Related to Why conservation of energy here vs momentum?

Why is conservation of energy important?

Conservation of energy is important because it is a fundamental law of nature that states that energy cannot be created or destroyed, only transferred or transformed. This means that energy is a limited resource and must be conserved in order to sustain our environment and way of life.

What is the difference between conservation of energy and conservation of momentum?

Conservation of energy and conservation of momentum are both fundamental laws of physics, but they apply to different aspects of motion. Conservation of energy refers to the total amount of energy in a system, while conservation of momentum refers to the total amount of momentum in a system. Energy can be transformed into different forms, but the total amount remains the same. Momentum is always conserved in a closed system, meaning it cannot be created or destroyed, only transferred between objects.

Why is momentum conserved in an isolated system?

Momentum is conserved in an isolated system because there are no external forces acting on the system. This means that the total amount of momentum in the system must remain constant, as there are no external forces to change it. In an isolated system, the total momentum before an interaction must equal the total momentum after the interaction.

How does conservation of energy play a role in renewable energy sources?

Conservation of energy is important in renewable energy sources because it ensures that energy is being used efficiently and sustainably. Renewable energy sources, such as wind and solar, rely on natural resources that are constantly replenished, but they still require energy conservation to be cost-effective and environmentally friendly. By conserving energy, we can reduce our reliance on non-renewable energy sources and preserve the Earth's resources for future generations.

Can conservation of energy and momentum be violated?

No, conservation of energy and momentum are fundamental laws of nature and cannot be violated. These laws have been extensively tested and proven to be true in countless experiments. However, there may be instances where it appears that these laws are being violated, but this is usually due to incomplete understanding of the system or external factors that were not taken into account.

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