motion#1
TRANSCRIPT
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Chapter 2
Motion in One Dimension
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Mechanics
Mechanics is a branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements also it deals with matter and investigates energy.Mechanics is divided into three branches:1- Statics2- Kinematics3- Dynamics
In this slide we will discuss KINEMATICS.
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Kinematics Kinematics is a part of mechanics that
studies motion in relationship to time. In kinematics, you are interested in the
description of motion Not concerned with the cause of the motion
(Dynamics) or bodies at rest (statics)
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Motion A body or object is said to be in motion when it is
observed moving or changing position relative to a fixed reference as time passes.
Therefore motion is relative; example when you are in a car and you take the moving car as your reference then you see a building moving back relative to you.
Then there should exist a reference frame to your work and since we are studying this topic while on Earth, we suppose the Earth is our reference frame thus it is called the terrestrial frame.
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Motion cont. Motion, to our level, is studied either according
to trajectory or according to speed. The trajectory is the path followed by the
center of mass of a moving object. Remark: our object could be: 1- either a solid body of considerable dimensions with a
center of mass where all the mass of the object is concentrated (also called center of gravity)
2- or a point mass with relative negligible dimensions and the mass is concentrated at the point itself.
Hint: don’t worry about this the question usually identifies the type of the object.
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Motion according to trajectory.
Studying the motion according to trajectory is very wide so we are limiting it to a general one which is curvilinear and includes one of three:
Rectilinear: along a straight line Circular: along a circle Curvilinear along a curve.The first study is the rectilinear motion
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Quantities in Motion Any motion involves three
concepts Displacement Velocity Acceleration
These concepts can be used to study objects in motion
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Position Defined in terms
of a frame of reference One dimensional,
so generally the x- or y-axis
Defines a starting point for the motion
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Displacement Defined as the change in position
or how much the object was displaced from its initial position.
f stands for final and i stands for initial May be represented as y if vertical Units are meters (m) in SI,
centimeters (cm) in cgs.
f ix x x
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Displacements
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Vector and Scalar Quantities Vector quantities need both
magnitude (size) and direction to completely describe them Generally denoted by boldfaced type
and an arrow over the letter + or – sign is also used in vector
representation. Scalar quantities are completely
described by magnitude only
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Displacement Isn’t Distance The displacement of an object is
not the same as the distance it travels Example: Throw a ball straight up and
then catch it at the same point you released it
The distance is twice the height The displacement is zero
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Speed The average speed of an object is
defined as the total distance traveled divided by the total time elapsed
Speed is a scalar quantity
total distanceAverage speed
total time
dv
t
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Speed, cont Average speed totally ignores any
variations in the object’s actual motion during the trip
The total distance and the total time are all that is important
SI unit m.s-1
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Velocity It takes time for an object to
undergo a displacement The average velocity is rate at
which the displacement occurs or simply the rate of change of displacement.
generally use a time interval, so let ti = 0
fi
averagefi
x xxv
t t t
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Velocity continued Direction will be the same as the
direction of the displacement (time interval is always positive) + or - is sufficient
Units of velocity is also m.s-1 (SI) Other units may be given in a
problem, but generally will need to be converted to m.s-1 these like km/h
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Speed vs. Velocity
Cars on both paths have the same average velocity since they had the same displacement in the same time interval
The car on the blue path will have a greater average speed since the distance it traveled is larger
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Graphical Interpretation of Velocity Velocity can be determined from a
position-time graph Average velocity equals the slope
of the line joining the initial and final positions
An object moving with a constant velocity will have a graph that is a straight line.
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Average Velocity, Constant
The straight line indicates constant velocity
The slope of the line is the value of the average velocity
Determine the equation of this straight line.
x= 2t - 20
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Average Velocity, Non Constant
The motion in this type of motion is of non-constant velocity
The average velocity is the slope of the blue line joining two points
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Instantaneous Velocity The limit of the average velocity as the
time interval becomes infinitesimally short, or as the time interval approaches zero
The instantaneous velocity indicates what is happening at every instant of time. It is a function of time.
The speedometer of a car indicates the instantaneous speed.
lim
0t
xv
t
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Instantaneous Velocity on a Graph The slope of the line tangent to the
position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as
the magnitude of the instantaneous velocity
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Instantaneous Velocity on a Graph
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Determine the instantaneous velocity at x = 15 m.About 2.5 ms-1
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Motion according to speed In this part we study the motion of an
object depending on its velocity along a rectilinear trajectory.
Our study is limited now to:1- motion with constant speed or velocity
and called uniform motion (URM)2- motion with a constantly varying
velocity and called uniformly accelerated motion. (UARM)
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Uniform rectilinear motion
URM is motion with constant velocity
The instantaneous velocities are always the same All the instantaneous velocities will
also equal the average velocity
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Kinematic equations of URM
a = 0 v = constant x = vt + xo
Where: a is acceleration v is velocity x is displacement xo is initial displacement or distance
covered when started timing
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Acceleration Changing velocity (not-uniform)
means an acceleration is present Acceleration is the rate of change
of the velocity
Units is ms-² (SI)
fi
fi
v vva
t t t
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Relationship Between Acceleration and Velocity
Uniform velocity (shown by red arrows maintaining the same size)
Acceleration equals zero
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Relationship Between Velocity and Acceleration
Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same
length) Velocity is increasing uniformly (red arrows are getting
longer) Positive velocity and positive acceleration, thus Accelerated
motion. va>0
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Relationship Between Velocity and Acceleration
Acceleration and velocity are in opposite directions
Acceleration is uniform (blue arrows maintain the same length)
Velocity is decreasing uniformly (red arrows are getting shorter)
Velocity is positive and acceleration is negative, thus decelerated motion. va<0
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Kinematic Equations Used in situations with uniform
acceleration a = const. 1
2
2 2
12122
v u at
x vt u v t
x ut at
v u a x04/11/23 IB Physics (IC NL) 31
2
3
4
5
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Kinematic equations cont. Equation 4 is derive when
plugging equation 2 in equation 3
Equation 5 is obtained when eliminating t from equation 2 and plug it in equation 3
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Notes on the equations
2i f
averagev v
x v t t
Gives displacement as a function of velocity and time.
Use when you don’t know and aren’t asked for the acceleration.
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Notes on the equations
Shows velocity as a function of acceleration and time
Use when you don’t know and aren’t asked to find the displacement
v u at
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Graphical Interpretation of the Equation
u
u
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Notes on the equations
Gives displacement as a function of time, velocity and acceleration
Use when you don’t know and aren’t asked to find the final velocity
21
2x ut at
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Displacement-time graph
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Acceleration graph When you are given a displacement-
time graph and the figure simulates a smiling face then the motion is accelerated. If it is a gloomy face then the motion is decelerated.
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Notes on the equations
Gives velocity as a function of acceleration and displacement
Use when you don’t know and aren’t asked for the time
2 2 2v u a x
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Graphical interpretation revisited
The area under the velocity-time graph represents the displacement.
Note: the area above the time axis is positive displacement where as the one below the axis Is negative.
Given the following diagram: the displacement is: x = 64 m
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Graphical interpretation revisited
The area under the acceleration-time graph represents the change in velocity.
Given the following graph: What is the velocity relative to this motion: v = 45 ms-1
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Problem-Solving Hints Read the problem Draw a diagram
Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations
Label all quantities, be sure all the units are consistent Convert if necessary
Specify the type of motion according to a. Choose the appropriate kinematic equation
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Problem-Solving Hints, cont Solve for the unknowns
You may have to solve two equations for two unknowns
Check your results Estimate and compare Check units
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Galileo Galilei 1564 - 1642 Galileo formulated
the laws that govern the motion of objects in free fall
Also looked at: Inclined planes Relative motion Thermometers Pendulum
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Free Fall All objects moving under the influence
of gravity only are said to be in free fall Free fall does not depend on the object’s
original motion All objects falling near the earth’s
surface fall with a constant acceleration The acceleration is called the
acceleration due to gravity, and indicated by g
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Laws of free fall First law: in vacuum and in the same
place, all bodies, dense or light obey the same law of fall.
Second law: the trajectory followed by a freely falling object is the vertical, directed towards the center of the earth.
Third law: the motion of a freely falling object is uniformly accelerated of acceleration g.
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Free fall on the moon
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Hyperlink for previous page
Apollo 15 Hammer/Feather gravity demonstration
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Motion in air Consequence: In air and in the same
place, dense bodies only and at the beginning of their fall obey the same law of fall as in vacuum.
When an object falls in air it is subjected to friction with air particles and to an opposing force called air resistance R.
R=kSV2
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Air resistance cont.
R = kSV2
R is air resistance k is the aerodynamic constant depending
on the shape of the object.S is cross-sectional area of object.V is velocity of object.
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Aerodynamic shape and k.
For each shape there is a different k.
The last shape F has the least k so that air resistance is minimal thus this shape is recommended for the use in airplanes and it is called the aerodynamic shape.
Note: the object is moving to the left.
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Terminal velocity When the air resistance reaches a
value that balances the weight of the object then
Ksv2=mg
This velocity is called the terminal velocity with which the object continues falling and it is constant.
ks
mgv
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Acceleration due to Gravity Symbolized by g g = 9.80 ms-²
When estimating, use g 10 m.s-2
g is always directed downward toward the center of the earth
Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion
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Free Fall – an object dropped Initial velocity is
zero Let up be positive Use the kinematic
equations Generally use y
instead of x since vertical
Acceleration is g = -9.80 m.s-2
u= 0
a = g
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Free Fall – an object thrown downward a = g = -9.80
m.s-2
Initial velocity 0 With upward
being positive, initial velocity will be negative
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Free Fall -- object thrown upward Initial velocity is
upward, so positive The instantaneous
velocity at the maximum height is zero
a = g = -9.80 m.s-2 everywhere in the motion
u = 0
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Thrown upward, cont. The motion may be symmetrical
Then tup = tdown
Then v = -u The motion may not be symmetrical
You can break the motion into various parts to understand but it is recommendable that you don’t do this and stick to one system of axes with specified directions that don’t change with the motion of the object.
Generally in initial direction of motion of the object.04/11/23 IB Physics (IC NL) 57
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Non-symmetrical Free Fall
No need to divide the motion into segments
Object thrown up: Vertical axis directed
upwards Horizontal axis through
projection point O. Above O, displacement is
positive Below O, displacement is
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Combination Motions
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