Download - Motions under non-constant acceleration
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Motions under non-constant acceleration
How do we know the position from the velocity function v(t)?
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Displacement from a Velocity - Time Graph
The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve
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Displacement during a time interval
First, we divide the time range from Ta to Tf into N equal intervals
N
TTt af
• Displacement xn during the n-th interval
tvxt
xv nn
nn
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Total displacement from Ta to Tf
N
nn
N
nn tvxx
11
In the limit as N approaches to infinite, the time interval t becomes smaller and smaller.
f
a
T
T
N
nn
t
dttvtvx )(10
lim
Displacement is the area under the curve of v(t).
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Position xf at Tf with initial position xa at Ta
f
a
T
Taf dttvxx )(
If the function v(t) is constant in time,
)( afaf TTvxx
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How do we know the velocity from the acceleration function a(t) ?
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Velocity vf at Tf with initial velocity va at Ta
f
a
T
Taf dttavv )(
If the function a(t) is constant in time,
)( afaf TTavv
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Position xf at Tf under a constant acceleration with initial velocity va at Ta=0
atvtv a )(
2
0
2
1
)(
ffaa
T
aaf
aTTvx
dtatvxxf
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Chapter 3
Motion in Two Dimensions
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3.1 Position and Displacement The position of an
object is described by its position vector,
The displacement of the object is defined as the change in its position
Fig 3.1
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Average Velocity The average velocity is
the ratio of the displacement to the time interval for the displacement
The direction of the average velocity is the direction of the displacement vector,
Fig 3.2
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Average Velocity, cont The average velocity between points is
independent of the path taken This is because it is dependent on the
displacement, which is also independent of the path
If a particle starts its motion at some point and returns to this point via any path, its average velocity is zero for this trip since its displacement is zero
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Instantaneous Velocity The instantaneous velocity is the limit of
the average velocity as ∆t approaches zero
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Instantaneous Velocity, cont The direction of the instantaneous
velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion
The magnitude of the instantaneous velocity vector is the speed
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Average Acceleration The average acceleration of a particle
as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.
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Average Acceleration, cont
As a particle moves, can be found in different ways
The average acceleration is a vector quantity directed along
v
Fig 3.3
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Instantaneous Acceleration The instantaneous acceleration is the
limit of the average acceleration as approaches zero
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Producing An Acceleration Various changes in a particle’s motion
may produce an acceleration The magnitude of the velocity vector may
change The direction of the velocity vector may
change Even if the magnitude remains constant
Both may change simultaneously
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3.2 Kinematic Equations for Two-Dimensional Motion When the two-dimensional motion has a
constant acceleration, a series of equations can be developed that describe the motion
These equations will be a generlization of those of one-dimensional kinematics to the vector form.
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Kinematic Equations, 2 Position vector
Velocity
Since acceleration is constant, we can also find an expression for the velocity as a function of time:
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Kinematic Equations, 3
The velocity vector can be represented by its components
is generally not along the direction of either or
Fig 3.4(a)
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Kinematic Equations, 4 The position vector can also be
expressed as a function of time:
This indicates that the position vector is the sum of three other vectors:
The initial position vector The displacement resulting from The displacement resulting from
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Kinematic Equations, 5
The vector representation of the position vector
is generally not in the same direction as or as
and are generally not in the same direction
Fig 3.4(b)
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Kinematic Equations, Components The equations for final velocity and final
position are vector equations, therefore they may also be written in component form
This shows that two-dimensional motion at constant acceleration is equivalent to two independent motions One motion in the x-direction and the other
in the y-direction
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Kinematic Equations, Component Equations becomes
vxf = vxi + axt and vyf = vyi + ayt
becomes
xf = xi + vxi t + 1/2 axt2 and yf = yi + vyi t + 1/2 ayt2
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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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