 Hello everyone, I am Swathi Ghargay, assistant professor, department of civil engineering from WIT, Sallapur. Topic for today's session is impact of elastic bodies. Learning outcome of this session, at the end of this session learner will be able to determine the velocity of each body after impact. Let us see first what is impact. It is a collision between two bodies. When the two bodies are in a contact for a small interval of time and exerts a very high pressure in this short time, it is called as the impact. On impact, body deforms first and then recover due to elastic properties and start moving with different velocity. So the velocity after impact is not only depend on the velocity of approach, but it is also depend on shape, size, elastic property and line of impact. In the figure one, it is shown the direct central impact. Here two bodies with moving along the line of impact. What is line of impact? Line of impact is the common normal of colliding surfaces. So they are moving on the line of impact. So it is called as the direct impact and why it is called as central impact? Because the center of masses of both the bodies are on the line of impact. So it is called as the direct central impact. Now see the figure number two. Here the two bodies are not moving on the line of impact. So they are moving in some other direction but the center of both the bodies are on the line of impact. Center means center of masses. So it is called as the oblique central impact. Oblique because they are not moving on the line of impact and central because the center of mass on the line of impact. Here the two bodies are moving along the line of impact but the center of mass of one body is above the line of impact. So it creates eccentricity. So it is called as the direct eccentric impact. Here refer figure number four now. In the figure number four, the one body which is not moving along the line of impact it is moving in some other direction and the mass center of another body is below the line of impact. So it is called as the oblique eccentric impact. Now we will see the period of collision. Period of collision it is the total time between the instant of initial contact and the instant of separation of the body. So the total time is called as the period of collision. Period of collision consists of two time interval, one period of deformation and second period of restitution. What is period of deformation? It is the time elapsed between the instant of initial contact and the instant of maximum deformation of the body and period of restitution is the time elapsed between the maximum deformation condition and the instant of separation of the body. See what happened exactly when the two bodies are in contact then it exerts a very high pressure on each other and it get deformed and due to the elastic property that it overcome that deformation and get separate and then move with some different velocity. Here the two body of mass m1 and m2 are moving with initial velocity u1 and u2. There is a collision between these two bodies and after impact they are moving with the different velocity let it be v1 and v2. Then the two bodies are in a contact they get deformed so the force exert during the period of deformation is Fd. So the impulse during the deformation is as we know that impulse is the force into time so force is Fd and dt is the small time interval in which it get deformed. So Fd into dt is the impulse during the deformation. After deformation it overcome the deformation and get separate. So this period is called as the period of restitution and in this period whatever the force act that force is called as the Fr force act during the period of restitution. So the impulse during the restitution is force into time that is Fr into dt and E is the coefficient of restitution. So the impulse during the period of restitution to the impulse during period of deformation the ratio of these two is called as the coefficient of restitution. As we know that impulse is the momentum and momentum is the product of mass and velocity and in particular that time interval velocity varies from the initial velocity to the final velocity. So the impulse is a final momentum minus initial momentum. So the coefficient of restitution we can write as v2 minus v1 divided by u1 minus u2. So what is v2 minus v1 it is the relative velocity of separation because this is the final velocity or we can say it is the velocity after impact. So the v2 minus v1 is the relative velocity of separation and u1 minus u2 it is the velocity of what is before impact. So this is called as the relative velocity of approach. So the coefficient of restitution is relative velocity of separation divided by relative velocity of approach. Now you pause the video here and answer the questions. Question 1 for perfectly elastic body coefficient of restitution will be your options are option A 1 option B 0 option C infinity option D not depend on elastic property of material of the body and question 2 is for perfectly in elastic body coefficient of restitution will be and your options are same 1 0 infinity and not depend on elastic property of material of the body. Here are the answer first before the answer of this two question you must know what properties of elastic material and what are the properties of in elastic material. So when the body is perfectly elastic it means that before impact and after impact they will move with the same velocity there will be a no loss in the velocity. So if there is no change in the velocity that ratio will be 1 when the body is perfectly in elastic it means that after impact both the body will not move because they are perfectly in elastic so they will not move they will steady at one position. So the velocity after impact will be 0 divided by some relative velocity of approach perfectly elastic in elastic body coefficient of restitution is 0. In this session we are learning the determination of velocity after impact. So after impact there are two velocities because there is a collision of two bodies so v1 and v2 we have to determine this two. So for that we need at least two equation so one equation we have that is coefficient of restitution and we need one more equation to find out this two unknown parameter. So we will take a help of the principle of conservation of momentum and principle of conservation of momentum is m1 u1 plus m2 u2 is equal to m1 v1 plus m2 v2. The same equation can be written in the form of weight also w1 by g into u1 plus w2 by g into u2 is equal to w1 by g into v1 plus w1 by g into v2. So g get cancelled because it is common everywhere so the same equation can be written as w1 u1 plus w2 u2 is equal to w1 v1 plus w2 v2. Now we will solve one numerical on the basis of equation of coefficient of restitution and law of conservation of momentum. The question is there is a direct central impact occur between 300 Newton body moving to the right with a velocity 6 meter per second and 150 Newton body moving to the left with a velocity 10 meter per second. And the velocity of each body after impact if the coefficient of restitution is 0.8. Here in the figure 6 the given data explain diagrammatically. So 300 Newton block is moving toward right with velocity 6 meter per second, 150 Newton body moving towards left with velocity 10 meter per second. So there is an impact between these two body and we have to determine the velocity after impact. Initially we do not know what will be the direction of velocity so there is an assumption both the bodies are moving towards the right. Now from the principle of conservation of momentum we will determine the first equation. So find out equation in the form of v1 and v2. So simplify the equation and it will be 2 v1 plus v2 is equal to 2. So mark it as an equation 1. Now use the definition of coefficient of restitution and derive second equation. It is v2 minus v1 is equal to 12.8. So these two are the equation 1 and 2 in that unknown are v1 and v2. So after solving it simultaneously we will get the v1 is equal to minus 3.6 meter per second and v2 is equal to 9.2 meter per second. These are my references thank you very much.