Momentum and Collisions

Transcription

1 Momentum and Collisions Vocabulary linear momemtum second law of motion isolated system elastic collision inelastic collision completly inelastic center of mass center of gravity 9-1 Momentum and Its Relation to Force Another quantity that is conserved is linear momentum and angular momentum and electric charge In this chapter we deal with more than one object linear momentum of an object is defined as the product of its mass and its velocity it is represented by p The relationship is given by the equation Equation Box 9-1 Since velocity is a vector then so is momentum the direction of momentum is the direction of velocity The unit of momentum is SI Units kg m/s- there is no special name for this unit The more momentum an object has the harder it is to stop it and the greater effect it will have if it is brought to rest by impact or collision A force is required to change the momentum of an object, whether it is to increase the momentum, to decrease it, or to change its direction The second law of motion- for system of objects- the rate of change of momentum of an object is equal to the net force applied to it We can write this as an equation Equation Box 9-2 Then is we take the relationship of momentum we obtain this equation 1 RoessBoss

2 Equation Box 9-3 It is important to know situations where the mass may change like rockets 9-2 Conservation of Momentum under certain circumstances momentum is a conserved quantity The sum of the momentum before and after are the same when it is conserved therefore we can write it as the momentum before = momentum after Equation Box 9-4 That is the total vector momentum of the system is conserved It stays constant It is closely connected to Newton s Laws of motion when the net external force on a system is zero, the total momentum remains constant The law of conservation of momentum- the total momentum of an isolated system of bodies remains constant an isolated system is one in which no external forces act- the only forces acting are those between objects of the system If a net external force acts on a system, then the law of conservation of momentum will not apply however if the system can be redefined so as to include the other objects exerting these forces, then the conservation of momentum principle can apply The importance of the momentum concept is that momentum is conserved under quite general conditions although the law of conservation of momentum follows from Newton s second law it is in fact more general than Newton s laws In the atom world- newtons laws fail- law of conservation, energy, angular momentum, and electric charge hold true 9-3 Collisions and Impulse In a collision of two ordinary objects, both objects are deformed, often considerable, because of the large forces involved 2 RoessBoss

3 Force usually jumps from zero at the moment of contact to a very large value within a very short tim and then returns abruptly to zero again Using Newtons Second Law F= p/t we can rearrange and then take it at an infinitesimal time we will obtain the following equation Equation Box 9-5 J is impulse- this is the integral of the force over the time interval Thus the change in the momentum is equal to the impulse acting on it The units for impulse are the same as momentum kg m/s (or N s) Impulse is the area under the time versus Force curve 9-4 Conservation of Energy and Momentum in Collisions during most collisions we usually do not know how the collision force varies over time, and so analysis using Newton s second law becomes difficult or impossible we can still determine a lot about the motion after a collision, given the initial motion, by making use of the conservation laws for momentum and energy as long as there are no external forces then it is conserved The moment that two objects contact one another there is interaction If all the momentum before and after is conserved then the interaction was an elastic collision- no sticking Equation Box 9-6 In the macroscopic world there is always some exchange of heat= in the atomic level there is not Even when the kinetic energy is not conserved, the total energy is of course always conserved Collisions in which kinetic energy is not conserved are said to be inelastic collisions The kinetic energy that is lost is changed into other forms of energy Equation Box RoessBoss

4 9-5 Elastic Collisions in one dimension We now apply the conservation laws for momentum and kinetic energy to an elastic collision between two small objects that collide head on, so all the motion in in line For conservation of momentum we have Equation Box 9-8 Because the collision is assumed to be elastic, kinetic energy is conserved Equation Box 9-9 In any elastic head on collision, the relative speed of the two objects after the collision has the same magnitude as before (but opposite direction), no matter what the masses are 9-6 Inelastic Collisions collisions in which kinetic energy is not conserved are called inelastic collisions some of the initial kinetic energy in such collisions is transformed into other types of energy, thermal or potential energy final kinetic energy is less than the initial energy the inverse can happen when potential energy such as chemical or nuclear is released Explosives are examples of this If two objects stick together as a result of the collision then this is considered inelastic collision Basically they couple together when they collide together On the side that they stick together we add the masses together 9-7 Collisions in Two or Three Dimensions conservation of momentum and energy can also be applied to collisions in two or three dimensions, and the vector nature of momentum is especially important we apply the same ideas with the vector rules just with all the dimensions 9-8 Center of Mass (CM) The point where rotation can occur is called the center of mass (CM) 4 RoessBoss

5 The general motion of an extended body (or system of bodies) can be considered as the sum of the translational motion of the CM plus rotational, vibrational, or other types of motion about the CM With rotation along the CM you get Rotation and Translational motion To determine the distance of the CM you can apply the following math Equation Box 9-10 The concept is similar to the center of mass and the center of gravity CG is that point at which the force of gravity can be considered to act Gravity acts on all the different parts or particles of a body, but for purposes of determining the translational motion of the body as a whole, we can assume the entire weight of the body For symmetrically shaped bodies- it is the geometric center 9-9 Center of Mass and Translational Motion The sum of all the forces acting on the system is equal to the total mass of the system times the acceleration of its center of mass (Newtons second law for a system) The center of mass of a system of particles (or bodies) with total mass M moves like a single particle of mass M acted upon by the same net external force 9-10 Systems of Variable Mass; Rocket Propulsion When it comes to masses that change- you apply calculus to the same formulas You have to take times to infinitesimally small units 5 RoessBoss