Chapter 6Momentum and CollisionsIntroduction Last ChapterW = F·d =∆Eno external (nc) force meansconservation of energy F = ma = m∆v/∆t = ∆(mv)/∆tno external force means “mv” isconserved. What is “mv”?Momentum The linear momentum of an object of mass m moving with a velocity is defined as the product of the mass and the velocity SI Units are kg m / s Vector quantity, the direction of the momentum is the same as the velocity’srpvrm=p vrrMore About Momentum Momentum components px= m vxand py= m vy Applies to two-dimensional motion Momentum is related to kinetic energy22pKEm=Impulse In order to change the momentum of an object, a force must be applied The time rate of change of momentum of an object is equal to the net force acting on it Gives an alternative statement of Newton’s second law( )f inetm v vt t−∆= =∆ ∆pFrrImpulse cont. When a single, constant force acts on the object, there is an impulsedelivered to the object is defined as the impulse Vector quantity, the direction is the same as the direction of the forcet= ∆I Fr rIrImpulse-Momentum Theorem The theorem states that the impulse acting on the object is equal to the change in momentum of the object If the force is not constant, use the average force applied= ∆ = ∆ = −r rrr rf it m mI F p v vAverage Force in Impulse The average force can be thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the intervalAverage Force cont. The impulse imparted by a force during the time interval ∆t is equal to the area under the force-time graph from the beginning to the end of the time interval Or, the impulse is equal to the average force multiplied by the time interval, avt∆ = ∆F prrExample - ImpulseExamplesExample – racquet tensionRacquetball goes over 190 mph, and tennis goes over 160 mph. How tight should you string your racquet to get to that speed?? Tighter better?Kung Fu punchKeep the fist out??Impulse Applied to Auto Collisions The most important factor is the collision time or the time it takes the person to come to a rest This will reduce the chance of dying in a car crash Ways to increase the time Seat belts Air bagsTypical Collision Values For a 75 kg person traveling at 27 m/s and coming to stop in 0.010 s F = -2.0 x 105N a = 280 g Almost certainly fatalSurvival Increase time Seat belt Restrain people so it takes more time for them to stop New time is about 0.15 secondsAir Bags The air bag increases the time of the collision It will also absorb some of the energy from the body It will spread out the area of contact Decreases the pressure Helps prevent penetration woundsExample – momentum / impulselightConservation of Momentum Momentum in an isolated system in which a collision occurs is conserved A collision may be the result of physical contact between two objects “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies An isolated system will have not external forcesConservation of Momentum, cont The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system remains constant in time Specifically, the total momentum before the collision will equal the total momentum after the collisionConservation of Momentum, Example The momentum of each object will change The total momentum of the system remains constantForces in a Collision The force with which object 1 acts on object 2 is equal and opposite to the force with which object 2 acts on object 1 Impulses are also equal and opposite For the whole system, I = 0 !!! Don’t have to worry about F or I.Conservation of Momentum, cont. Mathematically: Momentum is conserved for the system of objects The system includes all the objects interacting with each other Assumes only internal forces are acting during the collision Can be generalized to any number of objects1 1 2 2 1 1 2 2i i f fm m m m+ = +v v v vr r r rNotes About A System Remember conservation of momentum applies to the system You must define the isolated systemExamplesTypes of Collisions Momentum is conserved in any collision Inelastic collisions Kinetic energy is not conserved Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object Perfectly inelastic collisions occur when the objects stick together Not all of the KE is necessarily lostMore Types of Collisions Elastic collision Both momentum and kinetic energy are conserved Actual collisions Most collisions fall between elastic and perfectly inelastic collisionsInelastic Collisions When two objects stick together after the collision, they have undergone a perfectly inelastic collision Conservation of momentum becomesf21i22i11v)mm(vmvm+=+About Collisions Momentum is a vector quantity Direction is important Be sure to have the correct signsMore About Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns Solve the equations simultaneously2f222f112i222i11f22f11i22i11vm21vm21vm21vm21vmvmvmvm+=++=+Elastic Collisions, cont. A simpler equation can be used in place of the KE equation1 2 1 2i i f fv v ( v v )− = − −Examples - elasticElastic – 2 in 1 out possible?Summary of Types of Collisions In an elastic collision, both momentum and kinetic energy are conserved In an inelastic collision, momentum is conserved but kinetic energy is not In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the sameProblem Solving for One -Dimensional Collisions Coordinates: Set up a coordinate axis and define the velocities with respect to this axis It is convenient to make your axis coincide with one of the initial velocities Diagram: In your sketch, draw all the velocity vectors and label the velocities and the massesProblem Solving for One -Dimensional Collisions, 2 Conservation of Momentum:Write a general expression for the total momentum of the system before and after the collision Equate the