So one of the most fundamental and important laws in classical physics is the law of conservation of momentum now this law is applicable for microscopic as well as macroscopic objects, so this law can be described using the following statement so essentially if no external forces are acting on our system of objects well then the total momentum of our system of objects will remain constant so let's see exactly what that means by looking at the following example let's suppose we have objected one particle one with mass one traveling in the positive direction along the x-axis and its initial velocity is V one now at the same time a second particle with mass M two is traveling in the negative direction along the x-axis with initial velocity V two now eventually these two particles will collide, and they will bounce back and travel in opposite directions so the final velocity of object one and with mass m1 is v1 Prime and define a velocity after collision of object to our particle two with mass M two is v2 Prime now what this law of
conservation of momentum states is the following if we sum up the momenta of our objects before collision that sum will be exactly identical to the sum of the momenta after the collision and that's given by the following formula recall that momentum is simply mass times velocity of the object, so the momentum before the sum is equal to m1 times v1 plus m2 times v2 equals the momentum or the sum of the momentum after m1 v1 prime plus m2 v2 Prime now notice this sum is equal to a constant sum is equal to a constant and those two constants are exactly identical and that's exactly what the law states that our total momentum of the system of objects these two particles remains
constant now notice I can take these two terms and bring them both to the left side, and we get the following equation so this basically states that the change of momentum of object 1 plus the change in momentum of object 2 is equal to 0 and that's exactly what we have in this equation, so this is the law of conservation of momentum now let's try to derive this law from you the second law of motion recall that Newton's second law of motion is force or the net force acting on the object is equal to mass times acceleration which is equal to the derivative of our momentum function with respect to time now let's examine what actually takes place when our two particles collide well Newton's third law of motion tells us that particle 1 and particle to exert equal but opposite forces on one another, so this is Newton's third law of motion, so we can use this fact, and you can second law of motion to derive this equation, so if we sum up all the forces acting on our system of two objects we get the following son the force create, or the force exerted on an object 2 by object 1 minus the force exerted on object 1 by object 2 is equal to 0,so because this force has the same magnitude as this force it points in opposite direction because of Newton's third law of motion if I sum up these two forces since they point along the same axis this sum will equal to 0 now by using the second law of motion we can rewrite each of these forces using the following representation, so the force created by object 1 on object 2 is written as our derivative of our momentum of object 1 with respect to time and that's exactly what we do with this derivative of our objects two momentum with respect to time, so I can combine them in the following way and I notice I have momentum one - momentum two or change in momentum 1 - change in momentum 2 and that equals zero and that's exactly what we have, so this implies this the change in momentum of object 1 plus the change in momentum of object 2 is equal to zero and that's exactly identical to this equation so once again we derived the conservation of momentum law using really second law of motion up this implies that our p1 momentum of object 1 + momentum of object 2 is equal to a constant as long as we have an isolated system in which we have no external forces acting on our objects so let's look, and one application let's look in the following example let's suppose a 20,000 kilogram train with velocity 30 meters per second collides head-on with another car at rest with the same exact mass now if they move away together find a final velocity, so I have object 2 is stationary my object 1 is moving in the positive direction along the x-axis with initial velocity of 30 meters per second, and their collision they both move off with the same velocity, so we want to calculate what this velocity is, so we can use this equation where v1 Prime and v2 prime are the same exact velocity because they move away together so how do they collide it's as if they're one object so that means we can write the following formula mass of object one times velocity one plus mass of object two times velocity two is equal to the sum of their masses multiplied by the velocity so notice we can plot, or we can solve for velocity by bringing the mass and the sum of the masses over and that's exactly what we get so because V 2 is 0 we simply have m1 times v1 divided by the sum of their mass, and we get 20 thousand kilograms multiplied by 30 meters per second divided by the sum of their masses which is twice their mass is equal to 15 meters per second, so this is the velocity of the objects both of the objects after they collide
You must be logged in to post a comment.