Linear Momentum The linear momentum of a particle, or an Week 8: Chapter 9 object that can be modeled as a particle, of mass m moving with a velocity v is defined to Linear Momentum and Collisions be the product of the mass and velocity: pv m The terms momentum and linear momentum will be used interchangeably in the text Linear Momentum, cont Newton’ Law and Momentum Linear momentum is a vector quantity Newton’s Second Law can be used to relate Its direction is the same as the direction of the the momentum of a particle to the resultant velocity force acting on it The dimensions of momentum are ML/T ddvpdm v Famm The SI units of momentum are kg · m / s dt dt dt with constant mass Momentum can be expressed in component form: px = m vx py = m vy pz = m vz Conservation of Linear Momentum Conservation of Momentum, 2 Whenever two or more particles in an Conservation of momentum can be expressed mathematically in various ways isolated system interact, the total momentum p = p + p = constant of the system remains constant total 1 2 p1i + p= 2i p 1f + p 2f The momentum of the system is conserved, not In component form, the total momenta in each necessarily the momentum of an individual direction are independently conserved particle pix = pfx piy = pfy piz = pfz This also tells us that the total momentum of an Conservation of momentum can be applied to isolated system equals its initial momentum systems with any number of particles This law is the mathematical representation of the momentum version of the isolated system model 1 Conservation of Momentum, Archer Example Archer Example, 2 The archer is standing Conceptualize on a frictionless surface The arrow is fired one way and the archer recoils in the (ice) opposite direction Approaches: Categorize Newton’s Second Law – Momentum no, no information about Let the system be the archer with bow (particle 1) and the F or a arrow (particle 2) Energy approach – no, There are no external forces in the x-direction, so it is no information about isolated in terms of momentum in the x-direction work or energy Analyze Momentum – yes Total momentum before releasing the arrow is 0 Archer Example, 3 Impulse and Momentum dp Analyze, cont. From Newton’s Second Law, F dt The total momentum after releasing the arrow is Solving for d p gives ddtpF pp0 12ff Integrating to find the change in momentum Finalize over some time interval The final velocity of the archer is negative tf pp p Fdt I Indicates he moves in a direction opposite the arrow fit i Archer has much higher mass than arrow, so velocity is much lower The integral is called the impulse, , of the force acting on an object over t I Impulse-Momentum Theorem More About Impulse This equation expresses the impulse- Impulse is a vector quantity The magnitude of the momentum theorem: The impulse of the impulse is equal to the area force acting on a particle equals the change under the force-time curve The force may vary with in the momentum of the particle time pI Dimensions of impulse are M L / T This is equivalent to Newton’s Second Law Impulse is not a property of the particle, but a measure of the change in momentum of the particle 2 Impulse-Momentum: Crash Impulse, Final Test Example The impulse can also Categorize be found by using the Assume force exerted by time averaged force wall is large compared with other forces IFt Gravitational and normal This would give the forces are perpendicular same impulse as the and so do not effect the time-varying force does horizontal momentum Can apply impulse approximation Collisions – Example 1 Collisions – Example 2 Collisions may be the The collision need not result of direct contact include physical contact The impulsive forces between the objects may vary in time in There are still forces complicated ways between the particles This force is internal to This type of collision can the system be analyzed in the same Observe the variations in way as those that include the active figure physical contact Momentum is conserved Types of Collisions Collisions, cont In an elastic collision, momentum and kinetic In an inelastic collision, some kinetic energy energy are conserved is lost, but the objects do not stick together Perfectly elastic collisions occur on a microscopic level In macroscopic collisions, only approximately elastic Elastic and perfectly inelastic collisions are collisions actually occur limiting cases, most actual collisions fall in Generally some energy is lost to deformation, sound, etc. between these two types In an inelastic collision, kinetic energy is not conserved, although momentum is still conserved Momentum is conserved in all collisions If the objects stick together after the collision, it is a perfectly inelastic collision 3 Clicker Question Perfectly Inelastic Collisions In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is Since the objects stick necessary so that the final kinetic energy of the system together, they share the is zero after the collision? same velocity after the collision A. It is not possible mm11vvii 2 2 mm 1 2 v f B. The objects must have momenta with the same magnitude but opposite directions. C. The objects must have the same mass. D. The objects must have the same velocity. E. The objects must have the same speed, with velocity vectors in opposite directions. Elastic Collisions Elastic Collisions, cont Both momentum and Typically, there are two unknowns to solve for and so you kinetic energy are need two equations The kinetic energy equation can be difficult to use conserved With some algebraic manipulation, a different equation can be used mm11vvii 2 2 v –v = v + v mmvv 1i 2i 1f 2f 11ff 2 2 This equation, along with conservation of momentum, can be 11 used to solve for the two unknowns mmvv22 2211ii 2 2 It can only be used with a one-dimensional, elastic collision between two objects 11 mmvv22 2211ff 2 2 Collision Example – Ballistic Elastic Collisions, final Pendulum Example of some special cases Conceptualize Observe diagram m1 = m2 – the particles exchange velocities Categorize When a very heavy particle collides head-on with a very Isolated system of projectile and light one initially at rest, the heavy particle continues in block motion unaltered and the light particle rebounds with a Perfectly inelastic collision – the speed of about twice the initial speed of the heavy particle bullet is embedded in the block of wood When a very light particle collides head-on with a very heavy particle initially at rest, the light particle has its Momentum equation will have velocity reversed and the heavy particle remains two unknowns Use conservation of energy from approximately at rest the pendulum to find the velocity just after the collision Then you can find the speed of the bullet 4 Two-Dimensional Collision, Two-Dimensional Collisions example The momentum is conserved in all directions Particle 1 is moving at Use subscripts for velocity v 1 i and particle Identifying the object 2 is at rest Indicating initial or final values In the x-direction, the The velocity components initial momentum is If the collision is elastic, use conservation of kinetic m1v1i energy as a second equation In the y-direction, the Remember, the simpler equation can only be used for one- initial momentum is 0 dimensional situations Two-Dimensional Collision, Clicker Question example cont What is the direction of After the collision, the motion ofv m after the momentum in the x-direction 1i 2 is m v cos m v cos collision? 1 1f 2 2f After the collision, the A. up-left momentum in the y-direction B. up-right is m1v1f sin m2v2f sin C. down-left If the collision is elastic, apply the kinetic energy equation D. down-right This is an example of a E. Right only glancing collision Two-Dimensional Collision Example The Center of Mass Conceptualize There is a special point in a system or object, See picture called the center of mass, that moves as if Choose East to be the positive x-direction and all of the mass of the system is concentrated North to be the positive y- at that point direction Categorize The system will move as if an external force Ignore friction were applied to a single particle of mass M Model the cars as particles located at the center of mass The collision is perfectly inelastic M is the total mass of the system The cars stick together 5 Center of Mass, Extended Center of Mass, Coordinates Object The coordinates of the Similar analysis can be center of mass are done for an extended mxii x i object CM M Consider the extended my ii object as a system y i CM M containing a large mzii number of particles i zCM M Since particle separation M is the total mass of the is very small, it can be system considered to have a Use the active figure to observe effect of different constant mass masses and positions distribution Finding Center of Mass, Center of Mass, position Irregularly Shaped Object The center of mass in three dimensions can Suspend the object from one point be located by its position vector, rCM For a system of particles, The suspend from 1 another point rr m CMM i i i The intersection of the r is the position of the ith particle, defined by resulting lines is the i ˆˆˆ center of mass rijkiixyz i i For an extended object, 1 rr dm CM M Center of Gravity Center of Mass, Rod Each small mass element of an extended Conceptualize Find the center