Tru Physics

Elastic Collisions

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Tru Physics
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Introduction

In physics, an elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies remains the same. In an idealized elastic collision, there is no net conversion of kinetic energy into other forms such as heat, noise, or potential energy.

Fundamental Equations

In one-dimensional elastic collisions, the conservation of momentum and kinetic energy lead to two fundamental equations:

  1. Conservation of momentum:

m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}

  1. Conservation of kinetic energy:

\dfrac{1}{2} m_1 v_{1i}^2 + \dfrac{1}{2} m_2 v_{2i}^2 = \dfrac{1}{2} m_1 v_{1f}^2 + \dfrac{1}{2} m_2 v_{2f}^2

where m_1 and m_2 are the masses of the bodies, v_{1i} and v_{2i} are their initial velocities, and v_{1f} and v_{2f} are their final velocities.

Coefficient of Restitution

The coefficient of restitution (e) is a measure of the “elasticity” of a collision. For an elastic collision, e=1. This coefficient can be determined from the ratio of the initial and final relative velocities of the two bodies:

e = \dfrac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}

Applications

Elastic collisions are an idealized model, but they can be a useful approximation for systems where kinetic energy is conserved. Examples include collisions of hard spheres, atoms in a gas, or even galaxies (since the stars within them typically do not collide with each other, the total kinetic energy is approximately conserved).

Conclusion

While elastic collisions are an idealization, they play a critical role in understanding more complex, real-world collisions. Understanding these collisions is key in fields as diverse as mechanical engineering, materials science, and astrophysics.

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