3.3 Projectile Motion

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3.3 Projectile Motion The motion of an object under the

influence of gravity only The form of two-dimensional motion

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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

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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

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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

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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 i and vyi = vi sin i

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Projectile Motion – Velocity at any instant The velocity components for the

projectile at any time t are: vxf = vxi = vi cos i = constant

vyf = vyi – g t = vi sin i – g t

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Projectile Motion – Position Displacements

xf = vxi t = (vi cos i t yf = vyi t + 1/2ay t2 = (vi sinit - 1/2 gt2

Combining the equations gives:

This is in the form of y = ax – bx2 which is the standard form of a parabola

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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

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Projectile Motion Diagram

Fig 3.5

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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

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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:

sin

gi i

m

vt

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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

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More About the Range of a Projectile

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Range of a Projectile, final

The maximum range occurs at i = 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

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Non-Symmetric Projectile Motion

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Fig 3.10

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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

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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

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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

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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

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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

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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

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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.

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Total Acceleration, equations The tangential acceleration:

The radial acceleration:

The total acceleration:

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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

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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

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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’

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Fig 3.15(a)

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Fig 3.15(b)

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Exercises of chapter 3

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

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Chapter 31

Particle Physics

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31.1 Atoms as Elementary Particles

Atoms 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

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Discovery of New Particles 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

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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

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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

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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

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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

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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

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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

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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

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Forces and Mediating Particles

3.3 Projectile Motion

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