2. Chapter 8
Momentum

3. Goals for Chapter 8 To study momentum.
To understand conservation of momentum.
To study momentum changes during collisions.
To add time and study impulse.
To understand center of mass and how forces act on the c.o.m.
To apply momentum to rocket propulsion.

5. Definition of Total Momentum
The total momentum P of any number particles is equal to the vector sum of the momenta of the individual particles:
P = PA + PB + PC + …….
( total momentum of a system of particles)

6. Analysis of a collision

7. Conservation of Momentum
The total momentum of a system is constant whenever the vector sum of the external forces on the system is zero. In particular, the total momentum of an isolated system is constant.

8. An astronaut rescue

9. COLLISION

10. In collisions, we assume that external forces either sum to zero, or are small enough to be ignored. Hence, momentum is conserved in all collisions.
Pf = Pi

11. momentum AND kinetic energy are conserved.
Elastic Collisions
In an elastic collision,
momentum AND kinetic energy are conserved.
pf = pi
and
kf = ki

13. A 1 Dimensional collision problem

15. A 2 Dimensional collision problem

17. momentum AND kinetic energy are conserved.
Elastic Collisions
In an elastic collision,
momentum AND kinetic energy are conserved.
pf = pi
and
kf = ki

18. Completely Inelastic Collisions
In an inelastic collision, the momentum of a system is conserved,
pf = pi
but its kinetic energy is not,
Kf ≠ Ki
Completely Inelastic Collisions
When objects stick together after colliding, the collision is completely inelastic.
In completely inelastic collisions, the maximum amount of kinetic energy is lost.

21. USING BOTH CONSERVATION OF MOMENTUM AND CONSERVATION OF TOTAL ENERGY

22. Work, Kinetic Energy and Potential Energy
Kinetic energy is related to motion:
K = (1/2) mv2
Potential energy is stored:
Gravitational:
U = mgh
Spring:
U = (1/2)kx2
Conservation of total energy
Kf + Uf = Ki + Ui + Wother

23. The ballistic pendulum

24. momentum AND kinetic energy are conserved.
Elastic Collisions
In an elastic collision,
momentum AND kinetic energy are conserved.
pf = pi
and
kf = ki
For head-on collision of 2 objects:
vf,A = [(mB – mA)/(mA+ mB)]vi,A
vf,B = [(2mA)/(mA+ mB)]vi,A
Relative velocity: vf,B – vf,A = -(vi,B – vi,A)

26. Impulse for constant force

27. Impulse, duration of the impact

28. Impulse, duration of the impact

30. The center of mass of a system of masses is the point where the system can be balanced in a uniform gravitational field.

31. Center of Mass for Two Objects
Xcm = (m1x1 + m2x2)/(m1 + m2) = (m1x1 + m2x2)/M

33. Locating the Center of Mass
In an object of continuous, uniform mass distribution, the center of mass is located at the geometric center of the object. In some cases, this means that the center of mass is not located within the object.

35. Suppose we have several particles A, B, etc. , with masses mA, mB, …
Suppose we have several particles A, B, etc., with masses mA, mB, …. Let the coordinates of A be (xA, yA), let those of B be (xB, yB), and so on. We define the center of mass of the system as the point having coordinates (xcm,ycm) given by
xcm = (mAxA + mBxB + ……….)/(mA + mB + ………),
Ycm = (mAyA + mByB +……….)/(mA + mB + ………).

36. The velocity vcm of the center of mass of a collection of particles is the mass-weighed average of the velocities of the individual particles:
vcm = (mAvA + mBvB + ……….)/(mA + mB + ………).
In terms of components,
vcm,x = (mAvA,x + mBvB,x + ……….)/(mA + mB + ………),
vcm,y = (mAvA,y + mBvB,y + ……….)/(mA + mB + ………).

37. For a system of particles, the momentum P is the total mass M = mA + mB +…… times the velocity vcm of the center of mass:
Mvcm = mAvA + mBvB + ……… = P
It follows that, for an isolated system, in which the total momentum is constant the velocity of the center of mass is also constant.

38. Acceleration of the Center of Mass:
Let acm be the acceleration of the cener of mass (the rate of change of vcm with respect to time); then
Macm = mAaA + mBaB + ………
The right side of this equation is equal to the vector sum ΣF of all the forces acting on all the particles. We may classify each force as internal or external. The sum of forces on all the particles is then
ΣF = ΣFext + ΣFint = Macm