1 The Center of Mass

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7 Finding the Center of Mass by Integration (Omit)

8 Motion of the Center of Mass

9 Definitions of center of mass motion. Center of Mass Motion

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11 Conservation of Linear Momentum

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14 Kinetic Energy of a System

15 (Eq frequently used in rotational dynamics.) System Kinetic Energy:

16 Practice: Momentum and Kinetic Energy

Two masses move on a frictionless horizontal surface. M1 = 1kg, v1i = 4m/s. M2 = 2kg, v2i = 1m/s. a)Find the center of mass speed. b) The masses collide along a straight line. Find v1f if v2f = 2.3 m/s and no other external forces act.

18 c) Calculate the initial and final kinetic energies. Is the collision energetically possible? It is possible for kinetic energy to decrease due to the production of thermal energy in a collision.

19 Collisions and Impulse

20 Impulse is Area under F(t)

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22 Types of Collisions: ●Complete Inelastic: K  Thermal (v1f = v2f) ● Inelastic: K  Thermal (v1f ≠ v2f) ● Elastic: Ki = Kf (v1f ≠ v2f)

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24 The Center-of-Mass Reference Frame

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30 Problems

31 Practice: Collisions and Impulse

A bullet of 230 grains moves horizontally at 830 feet per second and strikes a 10lb wood block lying at rest on a horizontal surface. The bullet takes 1.0 millisecond to stop inside the block. a) Convert the data to SI units.

b) Calculate the speed the block moves at just after the bullet stops in the block. System momentum conserved when external impulse is negligible.

c) Calculate the kinetic energy of the bullet before the collision and of the moving block + bullet after the collision. What percent of the original kinetic energy is converted to other energies? What percent is retained as kinetic?

d) Calculate the impulse received by the block. e) If the collision lasts 1.0 millisecond, calculate the average force exerted on each object.

36 Practice

Two masses move on a frictionless horizontal surface. M1 = 1kg, v1i = 4m/s. M2 = 4kg, v2i = 1m/s in a laboratory. The masses collide elastically along a straight line. a) Show that in the center of mass frame that the initial velocities are +2m/s and -1m/s. (vcm = +2 from previous example) b) What are the final velocities in the lab frame?

c) Calculate the system-momentum before and after the collision in the lab-frame. d) Calculate the initial and final kinetic energies of the system in the lab-frame. Are these energies consistent with the definition of an ‘elastic collision’? This is consistent with an elastic collision

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Two objects collide in two dimensions. No external forces act at any time. In SI units: p1i = (3, 4) p2i = (2, 1) b) Make a sketch of the momentum vectors before and after the collision. a) If p1f = (3, 2), then calculate p2f. a) Pi = (3, 4) + (2, 1) = (3, 2) + (px, py) = Pf (5, 5) = (3+px, 2+py) px = 2 py =3 p2f = (2, 3) Pi Pf b)

41 c) Calculate the angle each momentum vector makes with the x-axis. b) d) Angle of final total momentum vector Pf = Pi = (5, 5): Pf

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