Coefficient Of Restitution And Impact Analysis

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Algorithms, Math, and Physics

Coefficient of restitution and impact analysis

Understanding collisions between objects requires the concept of the Coefficient of Restitution (COR). COR measures the elasticity of a collision, expressed as the ratio of the relative velocity of separation after impact to the relative velocity of approach before impact. It ranges from 0 for perfectly inelastic collisions, where objects stick together, to 1 for perfectly elastic collisions with no kinetic energy loss.

Mathematically, COR is defined as:

e = \frac{v_B^\prime - v_A^\prime}{v_A - v_B}

Here, v_A and v_B are the initial velocities of the two bodies, and v_A^\prime and v_B^\prime are their velocities after the collision.

To analyze such scenarios, I use Newton’s impact law and conservation of linear momentum. Newton’s law relates the relative velocities of the bodies, while momentum conservation ensures the total momentum before and after the collision remains constant. These principles lead to expressions for the post-collision velocities v_A^\prime and v_B^\prime.

For two bodies of masses m_A and m_B, the post-collision velocities are:

\begin{aligned} v_A^\prime & = \frac{m_A v_A + m_B v_B + m_B e (v_A - v_B)}{m_A + m_B} \\ v_B^\prime & = \frac{m_A v_A + m_B v_B - m_A e (v_A - v_B)}{m_A + m_B} \end{aligned}

The type of collision depends on the alignment of the mass centers and velocities of the bodies. In direct central impacts, both the mass centers and velocities lie along the line of action, making the analysis straightforward. In oblique central impacts, the velocities have components both along and perpendicular to the line of action. My analysis focuses on the line of action, where the impulse acts, and assumes that the velocity components perpendicular to it remain unchanged.

By understanding the COR and these impact classifications, I can determine the post-collision dynamics of bodies with precision. This is essential in various fields, from mechanical engineering to quantum physics.

For more insights into this topic, you can find the details here.