Do Now (9/23/13): Show that the kinetic energy of a particle of mass m is related to the magnitude of the momentum p of that particle by 2 p KE 2m Momentum Linear Momentum Momentum depends on the mass of an object and the speed it is going. p = mv Because velocity is a vector then momentum is, also. Units of momentum are kg-m/s Changes in Momentum The rate of change of momentum of a body is equal to the net force applied to it. ∑ F = ∆p / ∆t = (mv – mv0)/∆t = (m(v-v0))/∆t ∑ means “sum of” = m ∆v / ∆t = ma The force definition of change in momentum includes cases in which the mass may change (e.g., rockets) Impulse • Net force on an object is equal to the rate of change of momentum F = ∆p / ∆t • Therefore, we can define: Impulse = F ∆t = ∆p • A force acting for a length of time produces a change in momentum Impulse-Momentum Problems #1 Tennis ball (0.06 kg) travels at 10 m/s Hit by racket and goes 36 m/s in opposite direction What is change in momentum of ball? If ball is in contact with racket for 0.02 s, what is the average force applied? Impulse-Momentum Problems #1 Dp = m(vf - vi) = (0.06 kg)[36 m/s –(-10 m/s)] = 2.8 kg-m/s F = mDv/ Dt where Dv = 46 m/s = (0.06 kg)(46 m/s)/ (0.02 s) = 140 N Impulse-Momentum Problems #2 Pellet (0.001 kg) from gun travels at 100 m/s and embeds 0.02 m in target. What average force is exerted on the pellet by the target? Vav = (vf + vi)/2 = (0 m/s + 100 m/s)/2 = 50 m/s Dd = vDt therefore Dt = Dd/v = 0.02 m/50 m/s = 0.0004 s F = mDv/Dt = [(0.001 kg)(100 m/s)]/(0.0004 s) = 250 N Conservation of Momentum When two objects collide, the momentum before the collision must be equal to the momentum after the collision m1v1 + m2 v2 = m1v1' + m2v2' The total momentum of any group of objects remains the same unless outside forces act on the objects. The total momentum of an isolated system of bodies remains constant Collision Problems Elastic or inelastic Elastic: m1v1 + m2 v2 = m1v1' + m2v2’ where one or more of the v’s are usually zero Inelastic m1v1 + m2 v2 = (m1 + m2) v Collisions system momentum before a collision = system momentum after collision Two Kinds of Collisions Elastic Objects “bounce” off of each other without permanent changes Inelastic Objects “stick” to one another after collision May involve changes in shape, but not mass Elastic Collisions m v v m m m v v m m Elastic Collisions Transfer Momentum with No Loses Inelastic Collisions Initial Final m v m/s m m m P = mv v/2 m/s P = (2m)(v/2) = mv Momentum is a Vector—It has Direction! y x “y” value of momentum must add to zero after collision. Why? Total Momentum After a Collision Must Be Equal to Momentum After the Collision Notice that the small ball reverses direction after the collision. When this happens, the “negative” momentum of the small ball adds to the “positive” momentum of the large ball. Center of Mass The center of mass is the point in space that would move in the same path as a particle containing the mass of an object would if subjected to the same forces. For calculations, one can assume that all the mass of an object is concentrated at the center of mass In general the center of mass and the center of gravity are the same Path of CM is Path of a Projectile Locating the Center of Mass (CM) In a two particle system, the CM lies on a line joining the two masses x1 x x2 xCM xCM = m1x1 + m2x2 m1 + m2 Center of Mass in Car Crashes Can you use the concept of center of mass to explain why SUVs roll more easily than regular cars? Center of Mass in Car Crashes • • SUV is tall compared with its width At tip angle shown CM falls outside of vehicle body and it tips over • • Car is wider compared to height At tip angle shown CM falls inside of vehicle body and it does NOT tip over The Old Way of High Jumping— The Straddle Why is this easier than the straddle from a physics perspective? The Modern Fosbury Flop Flop vs Straddle Center of mass below bar Center of mass above bar Height of center of mass Height of center of mass Straddle • • • Flop Remember that: Potential Energy = mass x acceleration of gravity x height Less energy is required to raise the center of mass a smaller distance The flop requires less energy than the straddle Do Now (9/24/13): At an intersection, a 1500 kg car traveling east at 25 m/s collides with a 2500 kg van traveling north at 20 m/s. Find the velocity of the wreckage immediately after the collision. Practice: Complete the multiple choice questions on p. 168. Show work/reasoning in addition to picking a letter Brainstorm answers to conceptual questions