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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. 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 - PowerPoint PPT Presentation
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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