Section 5: Conservation of Momentum Student Objectives: •
Find the momentum of a system of moving or stationary objects. •
Use conservation of momentum to solve collision, impact or explosion problems. •
Recognize the difference between elastic and inelastic collisions. Concepts: Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum -­‐ it has its mass in motion. The amount of momentum that an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object. p = m • v One of the most powerful laws in physics is the law of momentum conservation. The law of momentum conservation can be stated as follows: For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2. An isolated system is one where no external forces are acting. We can see that momentum is conserved in this case by thinking about the relation between momentum and Newton’s laws. If the force on a system is zero, then we can see that Δp must be zero as well. Much like the conservation of energy law was can express this law as an equation stating that the momentum of a system does not change or is constant. Δp = 0 or, p = constant. 21 Collisions between objects are governed by laws of conservation of momentum and energy. When a collision occurs in an isolated system, the total momentum of the system of objects is conserved. Provided that there are no net external forces acting upon the objects, the momentum of all objects before the collision equals the momentum of all objects after the collision. If there are only two objects involved in the collision, then the momentum lost by one object equals the momentum gained by the other object. A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision. This energy could be converted into heat, sound or any other form of energy. Any macroscopic collision between objects will convert some of the kinetic energy into internal energy and other forms of energy, so no large scale impacts are perfectly elastic. Momentum is conserved in inelastic collisions, but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy. Worked Example: A car with a mass of 2,000 kg is travels at a speed of 50 m/s straight towards at an 18 wheeler truck. The truck has a mass of 15,000 kg. The trucker notices the car and breaks to a speed of 10 m/s before the car and truck hit head on. After the collision, the car and truck are stuck together (assume negligible loss of mass due to the collision). Find, a) the momentum of the system before the collision, b) the final speed of the truck and car, c) and how much energy was lost in the collision. a) We can start this problem in a similar fashion to any conservation problem. Here we can calculate the initial momentum of the truck and car system. The subscript “c” refers to the car and the subscript “t” refers to the truck. The total momentum of the system is given by, Since the car is moving in one direction and the truck is moving in the other, we can express that difference in direction using a minus sign, thus vc = −50 m/s and vt = 10 m/s. 22 b) Now that we have the total momentum, we can use conservation of momentum to find the final velocity of the car/truck wreck. After the collision the car and truck are moving as a single object and so we can find the momentum of the car/truck wreck by considering it as a single object with the mass of both the car and the tuck combined. c) This collision is an inelastic collision because there is energy lost in deforming the car and truck to stick together. In addition there will be energy lost in heat formed during the collision as well as other energy losses. The initial energy of the system is given by the kinetic energy of the car and the kinetic energy of the truck. The final energy is given by the kinetic energy of the car/truck wreck. The energy lost in the collision is the difference between the energy before and the energy after. 23 Homework: 1. A cannon with a mass of 200kg and a cannonball with a mass of 25kg are initially at rest. The cannon fires the cannonball forward with a speed of 30 m/s, find the recoil speed that the cannon rolls back with. 2. A 0.05 kg bullet is fired straight with a velocity of 150 m/s. The bullet hits and sticks in a block with a mass of 4 kg. a) Find the velocity of the bullet and block system immediately after the bullet hits and sticks in the block. b) Is this collision an elastic or inelastic collision? 24