Maximally Inelastic Collision

This topic covers Maximally Inelastic Collisions. claimed by apatel404

The Main Idea

A collision is a brief interaction between large forces. This could include two objects or several depending on the situation and how they collide is important. Collisions can be either inelastic,elastic, or maximally inelastic which is a subset of inelastic. Inelastic collisions occur when the object's kinetic energies are not conserved in the final and initial state. In maximally inelastic collisions, the objects in the system collide and stick together to form one object which has a new velocity and the mass of the object is the total mass of all the objects that have now combined into one.

A Mathematical Model

Maximally Inelastic Collisions can be based off the fundamental principle of momentum:

[math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net}{Δt} }[/math]

where p is the momentum of the system, F is the net force from the surroundings, Δt is the change in time for the process.

Using the principle of momentum, one can derive the final velocity of the object where the two initial objects have combined to become one since the interaction between the objects is brief so, [math]\displaystyle{ {Δt} ≈ {0} }[/math]. So rewriting the equation with the concept that the change in time is 0 yields:

[math]\displaystyle{ {ΔP_{system}} = {0} }[/math] where we can break the change in momentum of the system to it's initial and final components to get: [math]\displaystyle{ {P_{final}} = {P_{initial}} }[/math].

Now if we plug in the mass and velocity of object 1 and the mass and velocity of object 2 we see that: [math]\displaystyle{ m_1 v_1 + m_2 v_2 = \left( m_1 + m_2 \right) v \, }[/math] where v is the final velocity, which becomes

[math]\displaystyle{ v=\frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} }[/math]

Now if we apply the concept of conservation of energy:

[math]\displaystyle{ {E} = {Q} + {W} }[/math] where E is the total energy, Q is the heat given off, and W is the work done.

If we input the various types of energy in for the total energy such as kinetic, potential, and internal we get

[math]\displaystyle{ {ΔE_k}+{ΔE_p}+{ΔU}= {Q} + {W} }[/math] where ΔE_k is the change in kinetic energy,ΔE_p is the change in potential energy, and ΔU is the change in internal energy.

Based on the concept that the two objects have initial velocities and are going to combine into one, we can assume that the work done is negligible, the process is adiabatic,and the change in potential energy is negligible. The equation is simplified to:

[math]\displaystyle{ {ΔE_k} + {ΔU} = 0 }[/math] which we can break into the initial and final components to get: [math]\displaystyle{ {ΔU} = -({K_{final}} - ({K_{initial,1}} + {K_{initial,2}})) }[/math]

Kinetic Energy for objects that have a velocity smaller than the speed of light is defined as [math]\displaystyle{ \frac{1}{2}{mv^2} }[/math], so putting in the values for mass and velocity we get that:

[math]\displaystyle{ {ΔU} = \frac{1}{2}{(m_1 v_1 +m_2 v_2)^2}{m_1+m_2} - (\frac{1}{2}{m_1 (v_1)^2} + \frac{1}{2}{m_2 (v_2)^2}) }[/math]

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Connectedness

1. How is this topic connected to something that you are interested in?
2. How is it connected to your major?
3. Is there an interesting industrial application?

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

Internet resources on this topic

References

This section contains the the references you used while writing this page