1. 3.3 Projectile Motion
The motion of an object under the influence of gravity only
The form of two-dimensional motion

2. Assumptions of Projectile Motion
The free-fall acceleration is constant over the range of motion
And is directed downward
The effect of air friction is negligible
With these assumptions, the motion of the object will follow

3. Projectile Motion Vectors
The final position is the vector sum of the initial position, the displacement resulting from the initial velocity and that resulting from the acceleration
This path of the object is called the trajectory
Fig 3.6

4. Analyzing Projectile Motion
Consider the motion as the superposition of the motions in the x- and y-directions
Constant-velocity motion in the x direction
ax = 0
A free-fall motion in the y direction
ay = -g

5. Verifying the Parabolic Trajectory
Reference frame chosen
y is vertical with upward positive
Acceleration components
ay = -g and ax = 0
Initial velocity components
vxi = vi cos qi and vyi = vi sin qi

6. Projectile Motion – Velocity at any instant
The velocity components for the projectile at any time t are:
vxf = vxi = vi cos qi = constant
vyf = vyi – g t = vi sin qi – g t

7. Projectile Motion – Position
Displacements
xf = vxi t = (vi cos qi) t
yf = vyi t + 1/2ay t2 = (vi sinqi)t - 1/2 gt2
Combining the equations gives:
This is in the form of y = ax – bx2 which is the standard form of a parabola

8. What are the range and the maximum height of a projectile
The range, R, is the maximum horizontal distance of the projectile
The maximum height, h, is the vertical distance above the initial position that the projectile can reaches.
Fig 3.7

9. Projectile Motion Diagram
Fig 3.5

10. Projectile Motion – Implications
The y-component of the velocity is zero at the maximum height of the trajectory
The accleration stays the same throughout the trajectory

11. Height of a Projectile, equation
The maximum height of the projectile can be found in terms of the initial velocity vector:
The time to reach the maximum:

12. Range of a Projectile, equation
The range of a projectile can be expressed in terms of the initial velocity vector:
The time of flight = 2tm
This is valid only for symmetric trajectory

13. More About the Range of a Projectile

14. Range of a Projectile, final
The maximum range occurs at qi = 45o
Complementary angles will produce the same range
The maximum height will be different for the two angles
The times of the flight will be different for the two angles

15. Non-Symmetric Projectile Motion

21. Fig 3.10

30. 3.4 Uniform Circular Motion
Uniform circular motion occurs when an object moves in a circular path with a constant speed
An acceleration exists since the direction of the motion is changing
This change in velocity is related to an acceleration
The velocity vector is always tangent to the path of the object

31. Changing Velocity in Uniform Circular Motion
The change in the velocity vector is due to the change in direction
The vector diagram shows
Fig 3.11

32. Centripetal Acceleration
The acceleration is always perpendicular to the path of the motion
The acceleration always points toward the center of the circle of motion
This acceleration is called the centripetal acceleration
Centripetal means center-seeking

33. Centripetal Acceleration, cont
The magnitude of the centripetal acceleration vector is given by
The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion

34. Period
The period, T, is the time interval required for one complete revolution
The speed of the particle would be the circumference of the circle of motion divided by the period
Therefore, the period is

37. 3.5 Tangential Acceleration
The magnitude of the velocity could also be changing, as well as the direction
In this case, there would be a tangential acceleration

38. Total Acceleration
The tangential acceleration causes the change in the speed of the particle and is in the direction of velocity vector, which parallels to the line tangent to the path.
The radial acceleration comes from a change in the direction of the velocity vector and is perpendicular to the path.
At a given speed, the radial acceleration is large when the radius of curvature r is small and small when r is large.

39. Total Acceleration, equations
The tangential acceleration:
The radial acceleration:
The total acceleration:

40. 3.6 Relative Velocity
Two observers moving relative to each other generally do not agree on the outcome of an experiment
For example, the observer on the side of the road observes a different speed for the red car than does the observer in the blue car
Fig 3.13

41. Relative Velocity, generalized
Reference frame S is stationary
Reference frame S’ is moving
Define time t = 0 as that time when the origins coincide
Fig 3.14

42. Relative Velocity, equations
The positions as seen from the two reference frames are related through the velocity
The derivative of the position equation will give the velocity equation
This can also be expressed in terms of the observer O’

43. Fig 3.15(a)

47. Fig 3.15(b)

49. Exercises of chapter 3
2, 3,16, 20, 27, 30, 38, 46, 50, 54, 63

50. Chapter 31
Particle Physics

51. In this image from the NA49 experiment at CERN, hundreds of subatomic particles are created in the collision of high-energy nuclei with a lead target. The aim of the experiment is to create a quark-gluon plasma, in which the force that normally locks quarks within protons and neutrons is broken.

52. 31.1 Atoms as Elementary Particles
From the Greek for “indivisible”
Were once thought to be the elementary particles
Atom constituents
Proton, neutron, and electron
After 1932 (neutrons are found in this year) these were viewed as elementary for they are very stable
All matter was made up of these particles

53. Discovery of New Particles
Beginning in 1945, many new particles were discovered in experiments involving high-energy collisions
Characteristically unstable with short lifetimes ( from 10-6s to 10-23s)
Over 300 have been cataloged and form a particle zoo
A pattern was needed to understand all these new particles

54. Elementary Particles – Quarks
Now, physicists recognize that most particles are made up of quarks
Exceptions include photons, electrons and a few others
The quark model has reduced the array of particles to a manageable few
Protons and neutrons are not truly elementary, but are systems of tightly bound quarks

55. Fundamental Forces
All particles in nature are subject to four fundamental forces
Strong force
Electromagnetic force
Weak force
Gravitational force
This list is in order of decreasing strength

56. Nuclear Force Holds nucleons together
Strongest of all fundamental forces
Very short-ranged
Less than m (1fm)
Negligible for separations greater than this

57. Electromagnetic Force
Responsible for binding atoms and molecules together to form matter
About 10-2 times the strength of the nuclear force
A long-range force that decreases in strength as the inverse square of the separation between interacting particles

58. Weak Force
To account for the radioactive decay process such as beta decay in certain nuclei
Its strength is about 10-5 times that of the strong force
Short-range force
Scientists now believe the weak and electromagnetic forces are two manifestions of a single interaction, the electroweak force

59. Gravitational Force
A familiar force that holds the planets, stars and galaxies together
A long-range force
It is about times the strength of the nuclear force
Weakest of the four fundamental forces
Its effect on elementary particles is negligible

60. Explanation of Forces
Forces between particles are often described in terms of the exchange of field particles or quanta
The force is mediated by the field particles
Photons for the electromagnetic force
Gluons for the nuclear force
W+, W- and Z particles for the weak force
Gravitons for the gravitational force

61. Forces and Mediating Particles