Collision problems (Conservation of Momentum & Energy)

In summary, the conversation discusses the use of conservation of momentum and kinetic energy in solving collision problems, specifically perfectly inelastic collisions. The first problem presented involves a billiard ball collision, with one ball initially moving at 3.00 m/s and deflecting 30.0° from its original direction. The second ball is initially at rest and their masses are equal. The equations used are conservation of momentum and kinetic energy, but the latter leads to an unsolvable equation due to the unknown final velocities and direction angle. The key to solving this problem is to find the velocity relations between the two balls.
  • #1
Enharmonics
29
2
Hey all, first time poster here. I'm pretty confused about how exactly to use conservation of momentum and kinetic energy to work collision problems, specifically perfectly inelastic collisions (you could probably tell it was that kind, since K is conserved).

I'm actually stuck on two different questions, which is why I decided to post and see if someone here could maybe steer me in the right direction as to how to work these.

1. Homework Statement

A billiard ball traveling at 3.00 m/s collides perfectly elastically with an identical billiard ball initially at rest on the level table. The initially moving billiard ball deflects 30.0° from its original direction. What is the speed of the initially stationary billiard ball after the collision?

So I've got

##V_{1,i} = 3.00 \frac{m}{s} , V_{2,i} = 0 , m_1 = m_2 = m##

Homework Equations



##P_i = P_f , K_i = K_f##

The Attempt at a Solution



Because the initially moving billiard ball deflects 30.0°, I set up my conservation of momentum equation with an X and Y-component, so that

## P_x: m(3.00 \frac{m}{s}) + 0 = mV_{1,f}cos(30.0°) + mV_{2,f,x}##

and

## P_y: 0 = V_{1,f}sin(30.0°) - V_{2,f,y} ##

Of course, the m's in ##P_x## can cancel, since ##m_1 = m_2##. I don't know either the first or second ball's final velocities or even the direction angle at which the second ball ##m_2## moves after the collision. In the diagram I've drawn on my paper I just have the first ball moving along the x-axis and deflecting "upward" at an angle of 30.0°, with the second ball deflecting "downward" at some angle ##\theta##.

I tried using conservation of kinetic energy, but that just leaves me with

##\frac{1}{2}(3.00 \frac{m}{s})^2 + 0 = \frac{1}{2}V_{1,f} + \frac{1}{2}V_{2,f}##

which I can't solve for the target variable V_{2,f} because I only have V_{1,f} in terms of its components. I don't know what to do at this point.

[Mentor note: Second problem removed. Member asked to start a new thread for the second problem]
 
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  • #2
Enharmonics said:
I tried using conservation of kinetic energy, but that just leaves me with

12(3.00ms)2+0=12V1,f+12V2,f12(3.00ms)2+0=12V1,f+12V2,f\frac{1}{2}(3.00 \frac{m}{s})^2 + 0 = \frac{1}{2}V_{1,f} + \frac{1}{2}V_{2,f}

check the KE equation

in momentum conservation you have two equation relating the velocities; try to find out the velocity relations.
 

Related to Collision problems (Conservation of Momentum & Energy)

1. How do you calculate the conservation of momentum in a collision?

In a collision, the total momentum of the system before the collision is equal to the total momentum after the collision. This means that the sum of the momenta of all objects involved in the collision remains constant. To calculate the conservation of momentum, you can use the equation: Σpi = Σpf, where p represents momentum, i represents the initial state, and f represents the final state.

2. What is the difference between elastic and inelastic collisions?

Elastic collisions are those in which the total kinetic energy of the system is conserved. This means that the objects involved in the collision bounce off each other without any loss of energy. In contrast, inelastic collisions are those in which some of the kinetic energy is lost, usually in the form of heat or sound. Inelastic collisions are often seen in real-world situations, while elastic collisions are idealized scenarios.

3. How does the mass of objects affect the outcome of a collision?

The mass of objects involved in a collision affects the outcome in terms of momentum and kinetic energy. Objects with larger mass have a greater momentum, meaning they will exert a larger force on other objects in the collision. Additionally, objects with larger mass will have more kinetic energy, meaning they will have a greater ability to do work on other objects. However, the conservation of momentum and energy still apply, meaning the total mass before and after the collision will remain the same.

4. Can you use the conservation of momentum and energy to predict the outcome of a collision?

Yes, the conservation of momentum and energy can be used to predict the outcome of a collision. By analyzing the initial states of the objects and using the equations for conservation of momentum and energy, you can determine the final states of the objects involved in the collision. However, it is important to note that real-world collisions may not always be perfectly elastic or inelastic, so the predicted outcome may not always match the actual outcome.

5. How does the angle of collision affect the conservation of momentum and energy?

The angle of collision between objects does not affect the conservation of momentum and energy. This is because the laws of conservation apply in all directions, meaning the sum of the momenta and energies in the x, y, and z directions will remain constant. However, the angle of collision may affect the final velocities and directions of the objects after the collision, which can be calculated using the equations for conservation of momentum and energy.

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