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Lecture 12 Goals: • • Chapter 9: Momentum & Impulse Solve problems with 1D and 2D Collisions Solve problems having an impulse (Force vs. time) Chapter 10 Understand the relationship between motion and energy Define Potential & Kinetic Energy Develop and exploit conservation of energy principle Assignment: HW6 due Wednesday 3/3 For Tuesday: Read all of chapter 10 Physics 207: Lecture 12, Pg 1 Momentum Conservation FEXT dv d (mv) dP ma m dt dt dt and if FEXT 0 dP 0 implies that P constant dt P Momentum conservation (recasts Newton’s 2nd Law when net external F = 0) is an important principle (usually when forces act over a short time) It is a vector expression so must consider Px, Py and Pz if Fx (external) = 0 then Px is constant if Fy (external) = 0 then Py is constant if Fz (external) = 0 then Pz is constant Physics 207: Lecture 12, Pg 2 Inelastic collision in 1-D: Example A block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a final speed V. In terms of m, M, and V : What is the momentum of the bullet with speed v ? x v V before after Physics 207: Lecture 12, Pg 3 Inelastic collision in 1-D: Example What is the momentum of the bullet with speed v ? mv Key question: Is x-momentum conserved ? P After P Before mv M 0 (m M )V aaaa v V after x before v (1 M / m)V Physics 207: Lecture 12, Pg 4 Exercise Momentum is a Vector (!) quantity A. B. C. D. A block slides down a frictionless ramp and then falls and lands in a cart which then rolls horizontally without friction In regards to the block landing in the cart is momentum conserved? Yes No Yes & No Too little information given Physics 207: Lecture 12, Pg 5 Exercise Momentum is a Vector (!) quantity x-direction: No net force so Px is conserved. y-direction: Net force, interaction with the ground so depending on the system (i.e., do you include the Earth?) Py is not conserved (system is block and cart only) 2 kg 5.0 m 30° Let a 2 kg block start at rest on a 30° incline and slide vertically a distance 5.0 m and fall a distance 7.5 m 7.5 m into the 10 kg cart 10 kg What is the final velocity of the cart? Physics 207: Lecture 12, Pg 6 Exercise Momentum is a Vector (!) quantity 1) ai = g sin 30° = 5 m/s2 x-direction: No net force so Px is conserved y-direction: vy of the cart + block will be zero and we can ignore vy of the block when it 2) d = 5 m / sin 30° lands in the cart. j = ½ ai Dt2 N i 5.0 m 30° mg 30° Initial Final Px: MVx + mvx = (M+m) V’x M 0 + mvx = (M+m) V’x V’x = m vx / (M + m) = 2 (8.7)/ 12 m/s V’x = 1.4 m/s 10 m = 2.5 m/s2 Dt2 2s = Dt v = ai Dt = 10 m/s vx= v cos 30° = 8.7 m/s 7.5 m y x Physics 207: Lecture 12, Pg 7 A perfectly inelastic collision in 2-D Consider a collision in 2-D (cars crashing at a slippery intersection...no friction). V v1 q m1 + m2 m1 m2 v2 before after If no external force momentum is conserved. Momentum is a vector so px, py and pz Physics 207: Lecture 12, Pg 10 A perfectly inelastic collision in 2-D If no external force momentum is conserved. Momentum is a vector so px, py and pz are conseved V v1 q m1 + m2 m1 m2 v2 before after x-dir px : m1 v1 = (m1 + m2 ) V cos q y-dir py : m2 v2 = (m1 + m2 ) V sin q Physics 207: Lecture 12, Pg 11 Elastic Collisions Elastic means that the objects do not stick. There are many more possible outcomes but, if no external force, then momentum will always be conserved Start with a 1-D problem. Before After Physics 207: Lecture 12, Pg 12 Billiards Consider the case where one ball is initially at rest. after before pa q pb vcm Pa f F The final direction of the red ball will depend on where the balls hit. Physics 207: Lecture 12, Pg 13 Billiards: Without external forces, conservation of momentum (and energy Ch. 10 & 11) Conservation of Momentum x-dir Px : m vbefore = m vafter cos q + m Vafter cos f y-dir Py : 0 = m vafter sin q + m Vafter sin f after before pafter q pb F Pafter f Physics 207: Lecture 12, Pg 14 Force and Impulse (A variable force applied for a given time) Gravity: At small displacements a “constant” force t Springs often provide a linear force (-k x) towards its equilibrium position (Chapter 10) Collisions often involve a varying force F(t): 0 maximum 0 We can plot force vs time for a typical collision. The impulse, J, of the force is a vector defined as the integral of the force during the time of the collision. Physics 207: Lecture 12, Pg 15 Force and Impulse (A variable force applied for a given time) J a vector that reflects momentum transfer t t p J F dt (dp / dt )dt dp F Impulse J = area under this curve ! (Transfer of momentum !) t Dt Impulse has units of Newton-seconds ti tf Physics 207: Lecture 12, Pg 16 Force and Impulse Two different collisions can have the same impulse since J depends only on the momentum transfer, NOT the nature of the collision. F same area F Dt Dt big, F small t Dt t Dt small, F big Physics 207: Lecture 12, Pg 17 Average Force and Impulse F Fav F Fav Dt Dt big, Fav small t Dt t Dt small, Fav big Physics 207: Lecture 12, Pg 18 Exercise 2 Force & Impulse Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has the most momentum after the force acts ? F A. B. C. D. light F heavy heavier lighter same can’t tell Physics 207: Lecture 12, Pg 19 Boxing: Use Momentum and Impulse to estimate g “force” Physics 207: Lecture 12, Pg 20 Back of the envelope calculation t J F dt Favg Dt (1) marm~ 7 kg (2) varm~7 m/s (3) Impact time Dt ~ 0.01 s Question: Are these reasonable? Impulse J = Dp ~ marm varm ~ 49 kg m/s Favg ~ J/Dt ~ 4900 N (1) mhead ~ 6 kg ahead = F / mhead ~ 800 m/s2 ~ 80 g ! Enough to cause unconsciousness ~ 40% of a fatal blow Only a rough estimate! Physics 207: Lecture 12, Pg 21 Chapter 10: Energy (Forces over distance) We need to define an “isolated system” ? We need to define “conservative force” ? Recall, chapter 9, force acting for a period of time gives an impulse or a change (transfer) of momentum What if a force acts over a distance: Can we identify another useful quantity? Physics 207: Lecture 12, Pg 22 Lecture 12 Assignment: HW6 due Wednesday 3/3 For Tuesday: Read all of chapter 10 Physics 207: Lecture 12, Pg 23