In this video, I'm going to talk about energy loss during collision. I think this is a good ME question. As a mechanical engineer, you should relate how those energy loss during collision could be formulated. If I ask question to students like, "Is there momentum changes during collision or the energy changes during collision?" Most of the students could give me the right answer. If we defined the system as a two part colliding body, those impressive force are treated as internal force action-reaction pair. So yes, the momentum is conserved. However, when we are talking about the energy, depending on, it's a purely elastic collision or not, there is a heat occurs and so the energy is not conserved, where the case e equals not one. So the question is, why the e? This is the restitution coefficient, matters when we are dealing with the energy, e has nothing to do with the energy, seems like. E is a restitution coefficient and how those are affecting the energy conservation that's what we're going to talk about. Remember, recall that when we defined e, that's the restoration impulse divided by the deformation impulse. So whenever there is a two bodies in contact and maximum deformation and impending separation, those are the deformation period and then let's say the deformation displacement is delta xd. The time will be delta td. Same for the restoration. If we have a really accurate, precise measurement, may be they are slight difference, but let's assume that the deformation is going to be same as restoration, deformation is going to be assumed to be delta x and the time will be roughly evenly distributed for the deformation and restoration over the total collision time delta t. The question is then, is going to be deformation force equal to the restoration force. Well, if we think about a mass is attached to the spring, when there is a mass, there's a force is applied so that the spring has been compressed and bounced back. The total deformation force and restoration force will be equal, the symmetry, depending on the as a function of displacement. However, whenever there is a damping exist, those force might vary. If this non-linear damping and like the form is not just the one that we had built into vibration course like proportional to the velocity. Now, suppose this is a non-linear damping occurs, then it's hard to formulate them. So to make this simple, let's assume that, of course, those non-linearity or those damping friction due to the deformation will be applied throughout the whole collision period. But for simplification, let's assume all those friction and damping occurs during the restoration period. So there is a force by the restoration force and also force applied taken by the friction force. So this is how we assume moving out all the friction effect to the restoration side. So if that's our assumption, then deformation force will be at least equal or greater than the restoration forces depending on how big is the friction is. Is this concept everyone familiar? Well, actually you have this concept when you were in high school, when you are learning physics in high school. The restitution of coefficient e is defined by the restoration inverse divided by the deformation inverse, and assuming that those time durations are approximately similar, that is actually the ratio between the restoration force divided by the with respect to the deformation forces. So those e components has the force magnitude difference factored within. So Fr is going to be approximated as eFd. So all the collision, or friction, or damping effect are actually reflected in coefficient e. With that, we are going to apply work energy relationship, integrating F equals Ma over the displacement for mass number one and number two. During the deformation period, all the work done by the deformation force will generate the kinetic energy change for mass one and two. During the restoration period, in a similar manner, if you take the integral of the equations of motion over the displacement, restoration force work done is going to generate the kinetic energy for mass number one and number two. Then since we know that this Fr is going to be a function of Fd vectored by e on both sides, we can rewrite the equation like this, and if we sum them up, the total work done during collision is going to be work done by the deformation forces and the restoration forces for mass number one and mass number two. If you sum them up, these are the total work done during collision and these are the corresponding kinetic energy change. So to have energy to be conserved, what you have to do is, you should have a purely elastic collision, which means all those factored term of the friction and the damping turns out to be zero, which means e equals to one. Other than you have friction force exist, there should be work done. That means your energy is not conserved. This is how you could formulate the energy loss during collision, in terms of friction coefficient of restitution e.
