Collisions

Collision

The phenomenon in which the bodies moving independently interact for a very small interval of time exerting the impulsive forces on each other thereby changing the motion of each other is called collision. The forces in collisoin must be action-reaction pair and the linear momentum of the system must be conserved.

One dimensional elastic collision

In case of elastic collision, momentum as well as kinetic energy of the system remains conserved. From linear momentum conservation,\begin{align*} m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{align*}\begin{align*} m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})\end{align*} From KE conservation,\begin{align*} \frac{1}{2}m_{1}u_{1}^2+\frac{1}{2}m_{2}u_{2}^2=\frac{1}{2}m_{1}v_{1}^2+\frac{1}{2}m_{2}v_{2}^2\end{align*}\begin{align*} or, m_{1}(u_{1}-v_{1})(u_{1}+v_{1})=m_{2}(v_{2}-u_{2})(v_{2}+u_{2})\end{align*} Dividing the above equations, we get,\begin{align*} u_{1}+v_{1}=v_{2}+u_{2} \end{align*}\begin{align*} or,u_{1}-u_{2}=v_{2}-v_{1}\end{align*} ie. in case of elstic collision, velocity of approach is the same as the velocity of recession. We can write the initial velocity as \( v_{1}=v_{2}+u_{2}-u_{1} \). If we put the value of \(v_{1}\) in the above equation, we get,\begin{align*} v_{2}=\frac{2m_{1}u_{1}+(m_{2}-m_{1})u_{2}}{m_{1}+m_{2}}\end{align*} and also, we can get\begin{align*} v_{1}=\frac{2m_{2}u_{2}+(m_{1}-m_{2})u_{1}}{m_{1}+m_{2}}\end{align*} If the masses are same, colliding bodies exchange their velocities after collision.

One dimensional inelastic collision

The collision in which linear momentum is conserved but not the kinetic energy is caled inelastic collision. Consider, a system of two bodies having masses \(m_{1}\) and\(m_{2}\) moving with velocities\(u_{1}\) and\(u_{2}\) respectively as observed from the lab frame. Let, the bodies collide and coalesce. Let, \(v\) be the velocity of the system after collision. From linear momentum conservation,\begin{align*} m_{1}u_{1}+m_{2}u_{2}=(m_{1}+m_{2})v\end{align*}\begin{align*} or, v=\frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}\end{align*} KE before collision is,\begin{align*} E_{i}=\frac{1}{2}m_{1}u_{1}^2+\frac{1}{2}m_{2}u_{2}^2\end{align*} and, KE after collision is, \begin{align*} E_{f}=\frac{1}{2}(m_{1}+m_{2})v_{1}^2 \end{align*} If \(u_{2}=0\), \(E_{i}=\frac{1}{2}m_{1}u_{1}^2\) So, \begin{align*} \frac{E_{f}}{E_{i}}=\frac{1}{2}(m_{1}+m_{2})\frac{m_{1}^2u_{1}^2}{(m_{1}+m_{2})^2}\frac{1}{\frac{1}{2}m_{1}u_{1}^2}=\frac{m_{1}}{m_{1}+m_{2}}\end{align*}\begin{align*} \therefore \frac{E_{f}}{E_{i}}<1\end{align*} Hence, kinetic energy is not conserved in inelastic collision.

Kinetic Energy is conserved in center of mass frame of reference in an elastic collision.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.