Physics

1. Physics

2. Session System of Particle - 4

3. Session Objectives

4. Session Objectives • Collision-elastic and inelastic • Elastic collisions in one and two dimension • Coefficient of restitution • Explosions • Explosion in a projectile

5. M M M M M M M M M M M M m m m m m m m m m m m m Collision between two particles • Since Fext=0 so Pi=Pf • if KEi=KEf : Perfectly Elastic • if KEi > KEf : Inelastic All elastic collisions are perfectly elastic

6. Elastic collisons • Since Pi=Pf • Since KEi=KEf NOTE: Solving these two equations you can get the values of final velocities

7. Inelastic collisons • Since Pi=Pf • Since KEi >KEf NOTE: You need one more equality to solve for final velocities.

8. v2 v1 u2 u1 M M M m m m Coefficient of restitution Velocity of separation =e (velocity of approach) When e=1: elastic collision e=0: completely inelastic collision 0  e  1:Inelastic collision

9. General speed relations: (Remember)

10. Final speeds (special cases Elastics collision) (a) Heavy body and light body: (b) If m2>>m1(a light body hits a heavy body from behind) (c) Bodies of equal mass (m1=m2)

11. Questions

12. If a neutron with velocity v collides elastically with an  particle, what is its resultant velocity? Illustrative problem

13. Solution

14. u2 u1 v v M M M M m m m m Special case: perfectly inelastic collision (e=0)

15. Two Dimensional Collisions

16. Conservation of energy: Elastic collision in 2D Conservation of momentum: NOTE: There are 3 equations but 4 variables. So one physical condition is required to solve the problem.

17. 3m/s 4m/s  v Illustrative Problem A ball of mass 1kg moving with velocity V=5i collides with another ball of mass 2 kg initially at rest and at origin. The first ball comes to rest after collision and second breaks into two equal pieces. one piece starts moving with a velocity V=3j m/s. Then the velocity of other piece is:

18. Solution In x-direction 1x4+2x0 = 1x v cos In y-direction 0 = 1x3 –1xvsin Using given equation: tan=3/4 and v = 5m/s

19. Class Test

20. K m M Class Exercise - 1 A light spring of spring constant K = 1,000 N/m is kept compressed between two blocks of masses m = 10 kg and M = 15 kg on a smooth horizontal surface. When released the blocks acquire velocities in opposite directions. The spring loses contact with the blocks when it acquires natural length. If the spring was initially compressed through a distance x = 15 m, find the final speeds of the two blocks.

21. Solution When spring acquired its natural length, let the velocities of two blocks be v and u respectively. As no external force acts on the system,

22. Solution = 1.16 m/s

23. Two equal spheres A and B lie on a smooth horizontal circular groove at opposite ends of a diameter. A is projected along the groove and at time t it impinges on B. If e is coefficient of restitution, the second impact will occur after a time Class Exercise - 2

24. Let r be the radius of the groove, then arc AB = r • B A Solution Velocity of B before collision = 0 Velocity of A after collision = v Velocity of B after collision = v'

25. Solution Then v – v' = e(–u) Let t' be the time after next impact takes place, then

26. Class Exercise - 3 A neutron travelling with a velocity v and kinetic energy E collides elastically head-on with the nucleus of an atom of mass number A at rest. What fraction of total energy is retained by the neutron?

27. Solution 1 V = 1V1 + AVA Þ V2 – V12 = AVA2 Using (i),

28. Class Exercise - 4 A moving body with a mass m1 strikes a stationary body of mass m2. The masses m1 and m2 should be in the ratio m1/m2 so as to decrease the velocity of the first body 0.5 times assuming a perfectly elastic impact. What is the ratio of m1/m2 ?

29. Before Collision After Collision Solution Applying momentum conservation,

30. Solution Also,

31. Class Exercise - 5 A ball of mass m moving with a speed u undergoes a head on elastic collision with a ball of mass nm initially at rest. What fraction of the incident energy is transferred to the heavier ball?

32. Before Collision After Collision Solution Applying momentum conservation, Also, v2 – v1 = u

33. Class Exercise - 6 A body of mass m moving with a velocity v1 in the x-direction collides with another body of mass M moving in y-direction with a velocity v2. They coalesce into one body during collision. What is the direction and magnitude of the momentum of the final body?

34. Solution From conservation of momentum in x- and y-directions, we get

35. Two equal spheres A and B of masses 2 g and 30 g respectively lie at rest on a smooth floor, so that the line joining their centres is perpendicular to a fixed vertical wall, sphere A being nearer to the wall. A is projected towards B. Show that A is brought to rest after its second collision with B, if the coefficient of restitution between the two spheres and that between A and the wall is . Class Exercise - 7

36. Solution Let A be projected with a velocity u. Using (i) and (ii), we get Now, ball A moves towards the wall and returns back, say with a velocity VA’, then

37. Solution Solving (iii) and (iv), we get

38. A ball moving translation ally collides elastically with another, stationary ball of the same mass. At the moment of impact the angle between the straight line passing through the centers of the balls and the direction of the initial motion of the striking ball is equal to a = 45°. Assuming the balls to be smooth, find the fraction of the kinetic energy of the striking ball that turned into potential energy at the moment of the maximum deformation. Class Exercise - 8

39. y u a • x m m Solution Along X-axis, mu cosa = mv1x + mv2x When maximum deformation takes place, v1x = v2x u cosa = v1x + v2x = 2v1x

40. Solution

41. v3 mp 9 m/s 30o 30o Stationary ball v2 Class Exercise - 9 A ball moving with a speed of 9 m/s strikes an identical stationary ball such that after the collision, the direction of each ball makes an angle of 30° with the original line of motion. Find the speeds of the two balls after the collision if the kinetic energy is conserved in the collision process.

42. Solution Applying momentum of conservation in x- and y-directions, we get Also, in y-direction, Using equations (i) and (ii), we get

43. Solution Kinetic energy is not conserved.

44. 3 m/s 10 m/s m2=5 kg m1=2 kg Class Exercise - 10 Two blocks of masses m1 = 2 kg and m2 = 5 kg are moving in the same direction along a frictionless surface with speeds 10 m/s and 3 m/s respectively, m2 being ahead of m1. An ideal spring with K = 1,120 N/m is attached to the back side of m2. Find the maximum compression of the spring when the blocks collide.

45. 3 m/s 10 m/s m2=5 kg m1=2 kg Solution Let V be the velocity of the blocks of the time of maximum compression. Note that both the blocks will move with equal speeds at the time of maximum compression. Hence, From energy conservation,

46. Solution  100 – 65 = 560x2

47. Thank you