2. Physics 151: Lecture 18, Pg 1 Physics 151: Lecture 18 Today’s Agenda l Topics çReview of momentum conservation ç2-D Collisions çSystems of particles

3. Physics 151: Lecture 18, Pg 2 Lecture 17 ACT 4 The law of conservation of momentum applies to a collision between two bodies if: a.they exert forces on each other respectively proportional to their masses. b.they exert forces on each other respectively proportional to their velocities. d.their accelerations are proportional to their masses. e.they exert equal and opposite forces on each other.

4. Physics 151: Lecture 18, Pg 3 Lecture 18 Review problem: qualitative l Two boys in a canoe toss a baseball back and forth. What effect will this have on the canoe? Neglect (velocity-dependent) frictional forces with water or air. a. None, because the ball remains in the canoe. b. The canoe will drift in the direction of the boy who throws the ball harder each time. c. The canoe will drift in the direction of the boy who throws the ball with less force each time. d. The canoe will oscillate back and forth always moving opposite to the ball. e. The canoe will oscillate in the direction of the ball because the canoe and ball exert forces in opposite directions upon the person throwing the ball.

5. Physics 151: Lecture 18, Pg 4 Lecture 17, ACT 3 Momentum Conservation l Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. l The ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck. çWhich box ends up moving fastest ? (a) (b) (c) (a) Box 1 (b) Box 2 (c) same 1 2

6. Physics 151: Lecture 18, Pg 5 Lecture 17 Review problem l A 3.0-kg mass is sliding on a horizontal frictionless surface with a speed of 3.0 m/s when it collides with a 1.0-kg mass initially at rest as shown in the figure. The masses stick together and slide up a frictionless circular track of radius 0.40 m. To what maximum height, h, above the horizontal surface will the masses slide?

7. Physics 151: Lecture 18, Pg 6 Inelastic collision in 2-D l Consider a collision in 2-D (cars crashing at a slippery intersection...no friction). vv1vv1 vv2vv2 V beforeafter m1m1 m2m2 m 1 + m 2

8. Physics 151: Lecture 18, Pg 7 Inelastic collision in 2-D... l There are no net external forces acting. çUse momentum conservation for both components. vv1vv1 vv2vv2 V V = (V x,V y ) m1m1 m2m2 m 1 + m 2

9. Physics 151: Lecture 18, Pg 8 Inelastic collision in 2-D... l So we know all about the motion after the collision ! V V = (V x,V y ) VxVx VyVy  P pp1pp1 pp2pp2 P pp1pp1 pp2pp2 

10. Physics 151: Lecture 18, Pg 9 Elastic Collisions l Elastic means that energy is conserved as well as momentum. l This gives us more constraints. çWe can solve more complicated problems !! çBilliards (2-D collision). çThe colliding objects have separate motions after the collision as well as before. l Start with a simpler 1-D problem. BeforeAfter See text: 9.4

11. Physics 151: Lecture 18, Pg 10 Elastic Collision in 1-D v 1,b v 2,b before x m1m1 m2m2 See text: 9.4 v 1,a v 2,a after m1m1 m2m2

12. Physics 151: Lecture 18, Pg 11 Elastic Collision in 1-D v 1,b v 2,b v 1,a v 2,a before after x m1m1 m2m2 Conserve P X m 1 v 1,b + m 2 v 2,b = m 1 v 1,a + m 2 v 2,a Conserve Energy 1 / 2 m 1 v 2 1,b + 1 / 2 m 2 v 2 2,b = 1 / 2 m 1 v 2 1,a + 1 / 2 m 2 v 2 2,a Suppose we know v 1,b and v 2,b We need to solve for v 1,a and v 2,a Should be no problem 2 equations & 2 unknowns ! See text: 9.4 m 1 v 1,b + m 2 v 2,b = m 1 v 1,a + m 2 v 2,a

13. Physics 151: Lecture 18, Pg 12 Elastic Collision in 1-D l After some moderately tedious algebra, (see text book Chapter 9, section3) we can derive the following equations for the final velocities, See text: 9.4 1) m 1 v 1,b + m 2 v 2,b = m 1 v 1,a + m 2 v 2,a 2) v 1,b - v 2,b = - (v 1,a - v 2,a ) In general:

14. Physics 151: Lecture 18, Pg 13 Example - Elastic Collision l Suppose I have 2 identical bumper cars. One is motionless and the other is approaching it with velocity v 1. If they collide elastically, what is the final velocity of each car ? Note that this means, m 1 = m 2 = m v 2B = 0 See text: 9.4 Animation

15. Physics 151: Lecture 18, Pg 14 Lecture 18, ACT 3 Elastic Collisions l I have a line of 3 bumper cars all touching. A fourth car smashes into the others from behind. Is it possible to satisfy both conservation of energy and momentum if 2 cars are moving after the collision? All masses are identical, elastic collision. A) YesB) No Before After? Animation

16. Physics 151: Lecture 18, Pg 15 Example of 2-D elastic collisions: Billiards l If all we know is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things... See text: Ex. 9.11

17. Physics 151: Lecture 18, Pg 16 Billiards l Consider the case where one ball is initially at rest. ppappa ppbppb F PPaPPa beforeafter the final direction of the red ball will depend on where the balls hit. v v cm See Figure 12- 14 See text: Ex. 9.11

18. Physics 151: Lecture 18, Pg 17 ppP l We know momentum is conserved: p b = p a + P a l We also know that energy is conserved: l Comparing these two equations tells us that: Billiards pPpP p b 2 = (p a + P a ) 2 = p a 2 + P a 2 + 2 p a  P a m2 P m2 p m2 p 2 a 2 a 2 b  pP p a  P a = 0 and must therefore be orthogonal! Or … one momentum must be zero. ppappa ppbppb PPaPPa Ppp 2 a 2 a 2 b  See text: Ex. 9.11

19. Physics 151: Lecture 18, Pg 18 Billiards l The final directions are separated by 90 o. ppappa ppbppb F PPaPPa beforeafter v v cm See text: Ex. 9.11

20. Physics 151: Lecture 18, Pg 19 Lecture 18 – ACT 4 Pool Shark l Can I sink the red ball without scratching ? Ignore spin and friction. See text: Ex. 9.11 A) YesB) NoC) More info needed

21. Physics 151: Lecture 18, Pg 20 Lecture 18 – ACT 4 Pool Shark l From above, after the collision the two balls move off at right angles. l Thus if the red ball goes toward a pocket, so does the cue ball See text: Ex. 9.11 B) No

22. Physics 151: Lecture 18, Pg 21 Billiards. l More generally, we can sink the red ball without sinking the white ball – fortunately. See text: Ex. 9.11 Animation

23. Physics 151: Lecture 18, Pg 22 Recap of today’s lecture l Momentum and Collisions çCh. 9.1-9.4 (part of 9.4)