3. (a)
(9.1)
(a)
(9.2)
(b)
Figure 9-2 (a) Two particles of masses m1 and m2 are separated by distance d. The dot labeled com shows the position of the center of mass, calculated from Eq (b) The same as (a) except that the origin is located farther from the particles. The position of the center of mass is calculated from Eq. 9-2.The location of the center of mass with respect to the particles is the same in both cases.

5. (a) origin, (b)fourth quadrant, (c) on y axis below origin, (d) origin, (e) third quadrant, (f) origin

10. (a to c) at the center of the mass, still at the origin (their forces are internal to the system and can not move the center of mass of the system).

12. m1g + m2g + m3g = (m1 + m2 + m3) acom,
The motion of the center of mass of any system of particles is governed by Newton's second law, which is written as
Ʃ Fnet = M acom
For three balls of masses m1, m2 , and m3 , one can write
m1g + m2g + m3g = (m1 + m2 + m3) acom,
which gives
acom = g
Q. Three balls are thrown into the air simultaneously. What is the acceleration of their center of mass while they are in motion?
The center of mass moves like an imaginary particle of mass M under the influence of the resultant external force Fnet on the system.

14. (a) 1, 3, and then 2 and 4 tie (zero force); (b) 3

15. Comparing Eq with Eq. 9-17

18. (a) unchanged, (b) unchanged, (c) decrease

19. If the projectiles stop upon impact
(a) Zero, (b) positive (initial py down y, final py up y), (c) Positive direction of y.
If, the projectiles bounce directly backward from the target, with vf = -vi

24. No external force, p is conserved (a) 0, (b) no, (c) -x

25. In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum is constant. This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion. Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that F1 = dp1/dt and F2 = dp2/dt. Therefore
                 Or
                    
If the velocities of the particles are u1 and u2 before the interaction, and afterwards they are v1 and v2, then
                                    
This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.

26. Vpg = Vpw + Vwg
VHS = VHM + VMS

29. In General: if a massive particle and a light particle have the same momentum, the light one will have a lot more kinetic energy. If a light particle and a heavy one have the same velocity, the heavy one has more kinetic energy.

30. http://www. batesville. k12. in
(a) 10 kg.m/s; (b) 14kg.m/s and (c) 6 kg.m/s

31. the mechanical energy of the bullet–block–Earth
system is conserved:

36. Eq. 9-68
let’s start with the second collision in which block 2 stops because of its collision with block 3, Eq

38. (a) 2 kg. m/s (conserve momentum along x-axis, (b) 3 kg
(a) 2 kg.m/s (conserve momentum along x-axis, (b) 3 kg.m/s conserve momentum along y-axis)

42. the location of the automobile (of mass m1) is
While for the truck (of mass m2) is x2 = vt = (8.0 m/s)(3.0s) = 24 m.
(a) The location of their center of mass is

50. (a) From conservation of momentum, the momentum of bullet= to the momentum of bullet imbedded in the block