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

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

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

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

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

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

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

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

Projectile Motion Diagram Fig 3.5

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

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:

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

More About the Range of a Projectile

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

Non-Symmetric Projectile Motion

Fig 3.10

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

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

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

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

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

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

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.

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

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

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

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’

Fig 3.15(a)

Fig 3.15(b)

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

Chapter 31 Particle Physics

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.

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

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

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

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

Nuclear Force Holds nucleons together Strongest of all fundamental forces Very short-ranged Less than 10-15 m (1fm) Negligible for separations greater than this

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

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

Gravitational Force A familiar force that holds the planets, stars and galaxies together A long-range force It is about 10-41 times the strength of the nuclear force Weakest of the four fundamental forces Its effect on elementary particles is negligible

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

Forces and Mediating Particles