OpenStax College Physics for AP® Courses, Chapter 8, Problem 26 (Test Prep for AP® Courses)

This is College Physics Answers with Shaun Dychko. We have two cars—one labeled A and the other labeled B— of equal mass and they have an elastic collision and before the collision, car A is moving at positive 20 meters per second and car B is moving at negative 10 meters per second and assuming that both cars continue moving along the x-axis after the collision, what would be the velocities of each car after collision? So we know that the total momentum before the collision equals the total momentum after the collision and that is mass times v A plus mass times v B equals mass times v A prime plus mass times v B prime— the prime indicates after the collision— and the masses do not get subscripts A or B in this case because the cars have equal mass and that makes things easier because we can divide both sides by m and then we get this expression here: v A plus v B equals v A prime plus v B prime and I am going to write this as v A minus v A prime equals v B prime minus v B, we can subtract v B from both sides and we can subtract v A prime from both sides and it won't be obvious why I am doing this at this point but just take my word for it that it's a good idea and then let's consider the conservation of kinetic energy which we know occurs here because the collision is elastic so mechanical energy... or kinetic energy is maintained in this collision only because we are told specifically that it's an elastic one. So the total kinetic energy initially is one-half times mass times car A velocity squared plus one-half mv B squared and then that equals the total kinetic energy after collision one-half mv A prime squared plus one-half mv B prime squared and the one-half m can be canceled on both sides so divide both sides by one-half m and then you get this line here v A squared plus v B squared equals v A prime squared plus v B prime squared and I am going to instead write this as v A squared minus v A prime squared and that equals v B prime squared minus v B squared— same kinda trick that I did over here with the conservation of momentum business but here we are doing it again with conservation of kinetic energy— and then subtract v A prime squared both sides. Okay! And this is the difference of two things that are squared and there's a pattern for factoring that; we can write it instead as v A minus v A prime multiplied by v A plus v A prime and if you wanted to, you could multiply through and you would end up with v A squared minus v A prime squared. And likewise on the right hand side, factor in the difference of squares because v B prime minus v B times v B prime plus v B and the reason we have done all this business with subtracting things and moving terms around is so that after factoring, we can now divide this equation by this equation. We know that v A minus v A prime equals v B prime minus v B and so we can divide the left side by v A minus v A prime and then divide the right hand side by this, it looks different because it has letter B's for the subscript and terms have switched around but these things are equivalent so we can divide both sides by them and that's useful because now this... (oops... that's the wrong one) this disappears and this cancels over here leaving us with v A plus v A prime equals v B prime plus v B and this is only for elastic collisions between objects of equal mass. Okay! So now we are going to add equation one— this one here— to equation two and we'll get an answer finally So equation one says the left side is v A minus v A prime and left side equation two is v A plus v A prime and the right hand side for equation one is v B prime minus v B so we write that here and then the right hand side of equation two is v B prime plus v B and the v B's cancel here or they add to zero I guess you could say and these v A prime's add to zero and so we are left with 2v A on the left equals 2v B prime on the right or divide both sides by 2 and we find that v B prime equals v A. So whatever the initial velocity of car A was is now going to be the velocity of car B after the collision so car B sort of inherits the velocity of car A and it's going to be positive 20 meters per second and we can rearrange equation two and solve it for v A prime and it's going to be—after we subtract v A from both sides— we get that v A prime is v B prime plus v B minus v A. So that's 20 meters per second that we just found for v B prime plus the initial negative 10 meters per second for v B and then minus the initial velocity for v A, which is 20 meters per second and we end up with v A prime is negative 10 meters per second.