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Lecture 4
December, 2017, Lectures'::Edrees Harki
2
Motion :
The area of physics that we focus on is called mechanics: the study of the relationships between force, matter and motion
For now we focus on kinematics: the language used to describe motion
Later we will study dynamics: the relationship between motion and its causes (forces)
Simplest kind of motion: 1-D motion (along a straight line)
Be familiar with the following 2-D examples:
– projectile motion
– uniform and non-uniform circular motion
– general curve motion
December, 2017, Lectures'::Edrees Harki
Motion :1- Motion along a Straight Line
Simplest kind of motion: 1-D motion (along a straight line)
Motion is purely translational, when there is no rotation involved. Any object that is undergoing purely translational motion can be described as a point particle(an object with no size).
The position of a particle is a vector that points from the origin of a coordinate system to the location of the particle
The displacement of a particle over a given time interval is a vector that points from its initial position to its final position. It is the change in position of the particle.
Velocity and acceleration are physical quantities to describe the motion of particle
Velocity and acceleration are vectors For describe velocity and speed we must known the
distances and displacement.
December, 2017, Lectures'::Edrees Harki
Distance and Displacement Distance and displacement are two quantities
which may seem to mean the same thing, yet they have distinctly different meanings and definitions.
Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion.
Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's change in position.
Position and Displacement To study the motion, we need coordinate system
Motion of the “particle” on the dragster can be described in terms of the change in particle’s position over time interval
Displacement of particle is a vector pointing from P1 to P2 along the x-axis
Average velocity during this time interval is a vector quantity
whose x-component is the change in x divided by the time interval
t
x
tt
xxv xav
12
12
12 ttt 12 xxx
Average Speed and Velocity
Average speed is the total distance traveled divided by the time interval
Average velocity is the total displacement traveled divided by the time interval during which the displacement occurred
X-t Graph
This graph is pictorial way to represent how particle positionchanges in time
Average velocity depends onlyon total displacement x, not on the details of what happens during time interval t
The average speed of a particle is scalar quantity that is equal to the total distance traveled divided by the total time elapsed.
Instantaneous Velocity Instantaneous velocity of a particle is a vector equal to the limit
of the average velocity as the time interval approaches zero. It equals the instantaneous rate of change of position with respect to time.
dt
dx
t
xv
tx
0lim
On a graph of position as a function of time for one-dimensional motion, the instantaneous velocity at a point is equal to the slope of the tangent to the curve at that point.
Acceleration If the velocity of an object is changing with time, then the object is
undergoing an acceleration.
Acceleration is a measure of the rate of change of velocity with respect to time.
Acceleration is a vector quantity.
In straight-line motion its only non-zero component is along the axis along which the motion takes place.
Average Acceleration over a given time interval is defined as the change in velocity divided by the change in time.
In SI units acceleration has units of m/s2.
t
v
tt
vva xxx
xav
12
12
Instantaneous Acceleration▪ Instantaneous acceleration of an object is obtained by letting the
time interval in the above definition of average acceleration become very small. Specifically, the instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero:
dt
dv
t
va xx
tx
0lim
Constant Acceleration MotionIn the special case of constant acceleration:
the velocity will be a linear function of time, and
the position will be a quadratic function of time.
For this type of motion, the relationships between position, velocity and acceleration take on the simple forms :
12
12
tt
vva xx
x
0
0
t
vva xx
xtavv xxx 0
Position of a particle moving with constant acceleration
0
0
t
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vvv
tavtavvv xxxxxxav 2
1
2
1000
xxxav vvv 02
1
t
xxtav xx
00
2
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002
1tatvxx xx
Constant Acceleration Motion Relationship between position of a particle moving with
constant acceleration, and velocity and accelerationitself:
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tavv xxx 0x
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a
vvt 0
2
0000
2
1
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a
vva
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)(2
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)222(2
1
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1
))((
2
1
0
2
0
2
22
00
2
0
2
0
2
000
2
00
22
000
2
00
2
00
xxavv
vvaxx
vvvvvvva
xx
a
vvvvvv
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vvvva
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xxx
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xxxxxxx
x
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x
oxxx
tavv xxx 0
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1tatvxx xx
)(2 0
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Freely Falling Bodies
Example of motion with constant acceleration is acceleration of a body falling under influence of the earth’s gravitation
All bodies at a particular location fall with the same downward acceleration, regardless of their size and weight
Idealized motion free fall: we neglect earth rotation, decrease of acceleration with decreasing altitude, air effects
Galileo Galilei1564 - 1642
Aristotle
384 - 322 B.C.E.
Freely Falling Bodies The constant acceleration of a freely falling body is
called acceleration due to gravity, g
Approximate value near earth’s surface g = 9.8 m/s2
= 980 cm/s2 = 32 ft/s2
g is the magnitude of a vector, it is always positive
number
Free fall object experiences an acceleration of
g = 9.8 m/s2in the downward direction (toward the
center of the earth)
Define upwarddirection to be positive
Then a = -g = -9.8 m/s2
v = v0-g ty -y0= v0t -½ g t2
v2–v02= -2g ( y -y0)
y -y0= v t + ½ g t2
+
Constant Acceleration Motion
tavv xxx 0
2
002
1tatvxx xx
)(2 0
2
0
2 xxavv xxx
v = v0-g ty =y0+v0t - ½ g t2 ,
y -y0= v t + ½ g t2
V2 = v02 - 2g ( y -y0)
