Okay. Today we are going to study about special applications of kinetics of particle, F equals ma, and it's integral to the case for the impact, the collision case. So let's handle the direct collision, that the case, there are two bodies has the same direction of velocity case and the definition of coefficient of restitution. Suppose that there is a mass, m is moving with the velocity v1, which is greater than v2 of the velocity of mass 2, it'll ultimately will have a collision here and then if the body has it's elasticity, there will be a deformation occurs. Whenever there is a maximum deformation occurs, there will be a same velocity of the two body treated as the same body. A moment later there will be a restoration period occurs and so that its deformation will be restored. At some point there will be a separation occurs and then it'll have its after collision velocity, v1' and the v2'. So from the contact to the maximum deformation what we call this a deformation period and in between the mass 1 and mass 2, there will be a force, action-reaction force. So deformation force will be applied and then change the velocity from v1 to v0. From the restoration period, there will be another forces applied. Let's call it as a restoration forces F of r, and its direct action-reaction force to the mass 1 and mass 2 will change the velocity from v0 to v1' and v2'. If we are defining the whole system as mass 1 and 2, there is no external force. If we obtain the equations of motion, that's going to be F equals ma, which is zero. The linear momentum will be conserved. The initial linear momentum, m1v1, m2v2 is going to be equal to m1v1' and m2v2'. Since we have a two unknown that generally, the initial status of the system is given and we're usually supposed to find out what's going to happen after collision. Two unknowns, one equation, we still need more information. So here, we're going to define the e, the ratio of the restoration impulse versus deformation impulse as a coefficient of restitution e. So easy to memorize, deformation impulse on the denominator, d on the denominator. Will that be helpful? Now so by definition of the coefficient of restitution, let's derive the e from the mass 1. So for the deformation period, I have what impulse-momentum relationship. Initial momentum impulse and finer momentum and for the restoration period, initial momentum and linear impulse by the restoration force, and the final linear momentum. In Pi definition of v, which is going to be ratio for the restoration impulse to the deformation impulse, D on the denominator. I could express those impulse ratio in terms of: v0, v1, v1, and v0. Here, v0 is unknown and v1 is usually given and v1' is usually what we are supposed to find. Same procedures for the mass number 2 and I'm going to have impulse-momentum relationship for the deformation period and the restoration period. By the definition of e, restoration divided by the D, denominator, deformation impulse all I can have is all the information's about v2', v0, and v2. I have unknowns here, unknowns there, and then these two es are going to be equal. So I could have the relationship between the v2' and v1'. So I have two unknowns and I have two equations, one obtained from total systems impulse and momentum, second equation from the definition of the coefficient of restitution. I could specify v1' and v2' separately. So what if there is a collision has an angle? Oblique collision. So m1 has a velocity with respect to horizontal and with the amount of Theta 1 and two with the Theta 2. Whenever this collision occurs, there will be maximum deformation occurs and later it will be separated. So if I draw the free body diagram, there will be a large impulsive force back and forth. So it should be an action-reaction pair and generally, will be having some angular direction and then I can split it out a normal component and the tangential component. If I take the whole system, to bother the whole system, there's no external forces, so equations of motion will be F equal to 0 is going to be ma and therefore, the linear momentum will be conserved before and after. In the Cartesian coordinate, I can split it out as a component by component, as we mentioned earlier at the m equation video. So I have a normal component for the linear momentum will be conserved after collision and tangential components as well. So I have here four unknowns and only two equations. I still need more information. So let's work on the coefficient of restitution. From the collision, the initiation of contact to the maximum deformation, I defined it at deformation period. From maximum deformation to the separation period, I call it as a restoration period. The forces will be like within a shorter amount of time. Suppose this Delta Theta is really small and suppose that this magnet is really large, the force will have a sharp peak and release. If I obtained the impulse-momentum relationship for the mass number 1, I would have normal components of the initial linear momentum and impulse deformation force impulse, and the final linear momentum up to the maximum deformation period. Those turns out to be initial impulse and under restitution impulse and final linear momentum. By definition of e, restitution, restoration over the deformation, denominator deformation, I will have everything in terms of a v. Same for the mass number 2, I could obtain the e, restoration divided by the deformation at the denominator, and then I will have another relationship for the v' and if I make it as an equal, the same, what I could have is the e. I subscript and normal force because it's defined in the normal contact. There is relationships about v2' and v1' in the normal direction. Then there is another equations about v1' and v2' normal. So if I combine this together, I could specify what's going to be the normal speed for the mass number 1 and what's going to be the normal speed after collision for mass number 2. How about tangential? Can I use the same e defined for the tangential and obtain the tangential velocity? No. As I told you before, we assuming that. I haven't told you before, sorry, it's the first time. For the tangential, there's still is going to be a tangential force applied because there is a contact. However, the [inaudible] tangential force is really a lot smaller than the normal impulsive force. So most of the case to make the problem simple, a problem assumes that there is a very smooth surface, there is a contact. So the tangential impulse could be negligible. So since we can't define any deformation or restoration in tangential way, we cannot define the coefficient of restitution, e either. So here we learned about how we could handle the impact case, how we can apply impulse-momentum relationship, and also derive the coefficient of restitution, e, from the impulse-momentum relationship. Next video we are going to solve the example.