let's talk about the coefficient of restitution the coefficient of restitution is represented by the symbol e and it's equal to the ratio of the difference of the final velocities after the Collision to the initial velocities before the Collision so it's equal to V1 Prime minus V2 Prime / V2 minus V1 now the coefficient of restitution it's a number between zero and one when a coefficient of restitution equals 1 you're dealing with a perfectly elastic Collision now for all collisions momentum is conserve but for an elastic Collision the kinetic energy is conserved so there's no loss of kinetic energy for these types of collisions when the coefficient of restitution is equal to zero you have what is known as a completely inelastic collision for all types of collisions momentum is conserved for an inelastic collision kinetic energy is not conserved now if the coefficient of restitution is between zero and one you just have a regular inelastic collision so that's what the coefficient of restitution will tell you it will tell you if the Collision is elastic or if it's inelastic and to what degree is it inelastic is it completely inelastic or partially inelastic so now let's work on some practice problems let's start with this one a 5 kgr block moving East at 8 m/ second strikes a 10 kgr block at rest so let's draw a picture so let's say this is the 5 kgam block and it's moving East at 8 m per second and it's going to strike a 10 kilogram block and that block is at rest now after the Collision these two blocks they will be sticking together and in part A we need to find a combined final velocity of the two blocks so the initial momentum that is the total momentum before the Collision it's going to be M1 V1 plus M2 V2 and that's going to equal the final momentum which is going to be the sum of the two masses multiplied by their common final velocity now M1 is 5 kg V1 is 8 M2 is 10 V2 is Zer now M1 + M2 5 + 10 is 15 and we could solve for V final so 5 * 8 is 40 and that's equal to 15 VF so V final is going to be 40 / 15 that's 8 over3 which you could round that to 2.67 m/s so that's the answer for part A now Part B calculate the coefficient of restitution so that's going to be V1 primeus V2 Prime over V2 minus V1 so V1 Prime and V2 Prime the final velocities of the two blocks because they're sticking together they're moving at the same speed so V1 Prime and V2 Prime they're both equal to 2.67 now V2 was at rest so that was Zero V1 was 8 so because the two blocks are sticking together and they're moving at the same speed speed the coefficient of restitution will be zero because these two they will cancel out so what this tells us is that we're dealing with a completely inelastic collision so you're always going to have that situation whenever the two blocks stick together number two an 8 kilg block moving East at 6 m/s strikes a 4 kg block moving East at 2 m/s after the Collision the 8 kg block continues to move East at 4 m/s what is the final velocity of the 4 kg block after the Collision so here is the 8 kilogram block and it's moving East at 6 m per second and here we have another Block it's a 4 kilogram block but this one is moving East at 2 m/ second now after the Collision the 8 kilogram block is moving East but at a lower speed at 4 m/ second so lost some energy we need to find the final velocity of the 4 kg block and then we can move on the part B so this is V1 this is is V2 this is V1 Prime we need to find V2 Prime so we have M1 V1 plus M2 V2 that's going to equal M1 V1 prime plus M2 V2 Prime so M1 is 8 V1 is 6 M2 is 4 V2 is 2 M1 is 8 V1 Prime is 4 M2 is 4 and let's solve for v2 Prime so 8 * 6 is 48 4 * 2 is 8 8 * 4 is 32 and then 8 + 48 that's going to be 56 56 - 32 is 24 and divide on both s sides by 4 we have 24 over 4 which gives us 6 m/s so that's the final velocity of the 4 kgam block after the Collision notice that these two blocks they are not sticking together so because they have because after the Collision because they have different speeds and they're not sticking together this one will not be a completely inelastic collision but it will simply be just an inelastic collision so let's calculate the coefficient of restitution so it's going to be V1 Prime minus V2 Prime over V2 minus V1 so V1 Prime is 4 V2 Prime is 6 V2 is 2 and V1 is 6 so 4 - 6 is -2 2 - 6 is4 the negatives cancel 2 over 4 is5 so this is a value between 0o and one so that tells us that the Collision is simply an inelastic collision it's not an elastic Collision nor is it a completely inelastic collision it's simply an inelastic collision now we can see if we have an inelastic collision by checking if the kinetic energy has been conserved or not for an inelastic collision the kinetic energy is not conserved so the kinetic energy is going to be 12 mv^2 so for the first one it's going to be 12 8 * 6^ 2 half of 8 is 4 so 4 * 6 s that's 144 jewles now for the second one it's going to be 12 * 4 * 2^ 2 half of 4 is 2 2 * 2 2 that's 2 * 4 which is 8 for the next one it's 12 * 8 * 4^ 2 so that's going to be 64 jewles and for the last one half of 4 is 2 6 2 is 36 2 * 36 is is 72 so if we calculate the total before the Collision it's 144 + 8 which is 152 Jew and after the Collision it's 64 + 72 which is 136 Jew so we could see that we have a loss of -16 Jew of kinetic energy so because there was a loss of kinetic energy we know that we're dealing with an inelastic collision now if you were to follow the same procedure for the last problem the loss in kinetic energy is about 53.5 jewles when the coefficient of restitution is zero for this problem there was only -16 jewels of energy loss and the coefficient of restitution is 0.5 and when it's one we know that we're going to have a perfectly elastic Collision there will be no loss of kinetic energy so generally speaking based on the examples that we've seen in this video as the coefficient of restitution increases from 0 to 1 the loss of kinetic energy goes down as e goes up to one the loss of kinetic energy goes down to zero now let's work on this problem so we have a 4 kg ball and it's moving East at 5 m/s and is sh a 2 kg ball which is at rest now the 4 kg ball after the Collision it's going to move East at 1.67 m/s and a 2 kg ball is going to move East at a higher speed at 6.67 m per second calculate the coefficient of restitution so it's going to be V1 Prime minus V2 Prime over V2 minus V1 so this is V1 V2 is zero this is V1 Prime and this is V2 Prime so V1 Prime in this problem is 1.67 V2 Prime is 6.67 V2 the 2 kilogram balls at rest so that's zero and V1 that's going to be five so 1.67 - 6.67 that's -5 0 - 5 is5 which is one so because the coefficient of restitution is one what we have here is a perfectly elastic Collision so because we have an elastic Collision we know that the ktic energy is going to be conserved so there's no loss in kinetic energy and we can go ahead and determine that in this problem so if we calculate the kinetic energy for the first ball it's going to be 12 * 4 * 5^ 2 half of 4 is 2 5^ 2 is 25 2 * 25 is 50 now for the second ball it's not moving it's at rest so it has no kinetic energy so on the left we have 50 Jew of kinetic energy on the right this is going to be 12 * 4 * 1.67 2 and that's going to be about 5 578 jewles for the second one it's 12 * 2 * 6.67 squ so that's just 6.67 squ and that's going to be 4 44.4 89 jewles so if we add those two numbers together we get about 50 067 Jews now of course these are rounded figures so if we were to use exact answers we would get 50 Jews so for our practical purposes the change in kinetic energy is virtually zero so what we have here is an elastic collision and we can see that the coefficient of restitution is indeed one when you have an elastic Collision