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Mechanics Motion in One Dimension Ahmed H. Ali
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Motion in One Dimension
In this lesson we will discuss motion in one dimension. The oldest one of the
Physics subjects is the Mechanics that investigates the motion of a body. It deals
not only with a football, but also with the path of a spacecraft that goes from the
Earth to the Mars! The Mechanics can be divided into two parts: Kinematics and
Dynamics. The kinematics aims the motion. It is important for the kinematics that
which the path body follows. It answers the questions such that: Where the motion
started? Where the motion stopped? What time has taken for the complete of
motion? What the velocity body had?. The dynamics deals with the effects that
create the motion or change the motion or stop the motion. It takes into account the
forces and the properties of the body that can affect the motion. After that point,
we will enter into the world of kinematics, first. The One-Dimensional Motion is
the starting point for the kinematics. We will introduce some definitions like
displacement, velocity and acceleration, and derive equations of motion for bodies
moving in one-dimension with constant acceleration. We will also apply these
equations to the situation of a body moving under the influence of gravity alone.
As Leonardo Da Vinci said, “To understand motion is to understand nature”, we
need to understand the motion by observing and doing practice (experiment) on it,
first. The reason is quite simple. Those things that are of interest in science are the
things that undergo change. It is basic principle that: to understand how something
works, you have to see it in action. The workings of the universe include anything
in the universe which experiences change according to some repetitive pattern. It is
fact that change could be in the form of a chemical reaction, an increase or
decrease in the population of butterflies, etc. In nature, the easiest changes to
observe are those of motion: An object is moved from one position in space to
another. In fact, the motion generally leaves the object itself unchanged and thus
simplifies the observation. We first need some definitions to identify the motion. It
begins by defining the change in position of a particle. We call it “displacement”.
We first need some definitions to identify the motion. It begins by defining the
change in position of a particle. We call it “displacement”. Displacement is
defined to be the change in position or distance that an object has moved and is
given by the equation;
Mechanics Motion in One Dimension Ahmed H. Ali
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(1)
(The symbol Δ, the Greek uppercase delta, represents a change in a quantity, and it
means the final value of that quantity minus the initial value.)
Where is the final position and is the initial position. The arrow indicates
that displacement is a vector quantity: it has direction and magnitude, as we
mentioned before. In fact, is in 3-dimension and written as . In
our calculations, we will only take x-component of r into account. In 1-dimension,
there are only two possible directions that can be specified with either a plus (+) or
a minus (-) sign. As we know, other examples of vectors are velocity, acceleration
and force. In contrast, scalar quantities have only magnitude. Some examples of
scalars are speed, mass, temperature and energy.
It is not enough to define the displacement for a motion. In order to be useful, we
also need to specify something about time. After all, a motion that causes a
displacement of 1 meter can be large or small depending on whether it took a
second or a thousand years to do it. The international standard unit of time is the
second. Hence, time and space are inextricably linked in physics since we need
both to explain motion and motion is fundamental to all areas of physics. In the
language of mathematics, we describe the changes in position
, (2)
or time as
where the i and f subscripts depict initial and final, respectively. Generally, ti=
t0=0. There is no doubt that this use of symbols requires some patience if you are
not already comfortable with it. The normal tendency of students new to physics is
to immediately replace symbols with numbers as soon as possible. Experience will
show you that this is generally not a good thing to do. If we continue our rather
legal-sounding definitions, it is noted that there are two other aspects of motion
that are important. The first is that we would like to quantify the amount of motion
Mechanics Motion in One Dimension Ahmed H. Ali
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taking place. Therefore, we should define the average velocity or speed of an
object. As we said above, the idea of quantifying motion involves both the distance
traveled by the body and the time the body took to travel it. For that reason, it
makes since to define the average velocity that is the displacement over total
time, as follows:
(3)
When numbers are inserted for the position values x1 and x2 in Eq. 2 , a
displacement in the positive direction always comes out positive, and a
displacement in the opposite direction (left in the figure) always comes out
negative.
For example, if the particle moves from x1 = 5 m to x2 = 12 m, then the
displacement
is Δx = (12 m) - (5 m)=7 m. The positive result indicates that the motion is in the
positive direction. If, instead, the particle moves from x1 = 5 m to x2 = 1 m, then Δx
= (1 m) - (5 m) = - 4 m. The negative result indicates that the motion is in the
negative direction.
The actual number of meters covered for a trip is irrelevant; displacement involves
only the original and final positions. For example, if the particle moves from x = 5
m out to x =200 m and then back to x= 5 m, the displacement from start to finish is
Δx =(5 m) - (5 m) = 0.
A plus sign for a displacement need not be shown, but a minus sign must always be
shown. If we ignore the sign (and thus the direction) of a displacement, we are left
with the magnitude (or absolute value) of the displacement. For example, a
displacement of Δx= -4 m has a magnitude of 4 m.
Displacement is an example of a vector quantity, which is a quantity that has
both a direction and a magnitude. We need is the idea that displacement has two
features: (1) Its magnitude is the distance (such as the number of meters) between
the original and final positions. (2) Its direction, from an original position to a final
position, can be represented by a plus sign or a minus sign if the motion is along a
single axis.
Mechanics Motion in One Dimension Ahmed H. Ali
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Graphical interpretation of velocity:
Consider 1-dimensional motion from point A (with coordinates xi, ti ) to point B at
(xf , tf ). We can plot the trajectory on a graph (see Figure)
Then from Eq. (3) is just the slope of the line joining A and B. We have to notice
that we only deal with the final and initial point. We know “nothing” about the
motion of the body that moves between point A and point B. We do not know what
path the body followed OR what the shape of the body is OR what kinds of forces
are applied on the body OR the body applied on surroundings? It means that you
wake up in the dormitory and have gone to class in the morning and returned to the
dormitory after 8 hours. Since the total displacement is “ZERO”, your average
velocity is “ZERO”
Let’s assume that the average velocity of the particle be different for different time
intervals. In that condition, the particle’s velocity will differ for each time interval
and it will be necessary to calculate its velocity for a given certain time, t. This
leads us to define instantaneous velocity. Since the particle may not follow a
straight line on its path, the displacement vectors of that particle will differ from
each one by direction and magnitude. If the displacement occurs in time t after a
certain time Δt, then the displacement will be Δr. If the displacement is
infinitesimal small, then the velocity will take a certain value for that time interval.
In Mathematics,
(4)
Mechanics Motion in One Dimension Ahmed H. Ali
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This is the instantaneous velocity of the particle and its magnitude is called speed.
Average acceleration is the change in velocity over the change in time:
(5)
The direction of the acceleration is in the direction of the vector Δv, and its
magnitude is | Δv/ Δt |.
As we have followed before, we can find the Instantaneous acceleration of the
body; Instantaneous acceleration is calculated by taking shorter and shorter time
intervals, i.e. taking : Δt→0
(6)
It should be noted that:
Acceleration is the rate of change of velocity. When velocity and acceleration are
in the same direction, speed increases with time. When velocity and acceleration
are in opposite directions, speed decreases with time. Graphical interpretation of
acceleration: On a graph of v versus t, the average acceleration between A and B is
the slope of the line between A and B, and the instantaneous acceleration at A is
the tangent to the curve at A. From now on “velocity” and “acceleration” will
refer to the instantaneous quantities.
One Dimensional Motion with Constant Acceleration
As it is understood from the subtitle, constant acceleration means velocity
increases or decreases at the same rate throughout the motion. Example: an object
falling near the earth's surface (neglecting air resistance).
Derivation of Kinematics Equations of Motion
We choose ti=0, xi=xo, vi=v0x, and tf=t, xf=x, vf=vx. Since =constant, then =.
Then Eq. (6) can be written as
, or
(7)
Mechanics Motion in One Dimension Ahmed H. Ali
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Since “a ” is constant, vx changes uniformly and
. From Eq. (3) we
know that
,. Now, we can combine
. and
can use Eq. (7)
We get
(8)
This is the equation of motion for a particle in 1-dimension, with a constant
acceleration.
If we rewrite Eq. (7) we find →
Then substituting
this result into the last term of Eq. (8) we find
(9)
It should be noted that this equation does not depend on time, t.
Freely Falling Bodies
We call that a freely falling object is an object that moves under the influence of
gravity only. By neglecting air resistance, all objects in free fall in the earth's
gravitational field have a constant acceleration that is directed towards the earth's
center, or perpendicular to the earth's surface, and of magnitude =g= 9.8m/sec2.If
motion is straight up and down and we can choose a coordinate system with the
positive y-axis pointing up and perpendicular to the earth's surface, then we can
describe the motion with Eq. (7), Eq. (8), Eq. (3.9) with .
(Negative sign arises because the coordinate system is changed and the
acceleration direction is downward.) .
So that, equations of motion for the 1-dimensional vertical motion of an object in
free-fall can be written as following:
Mechanics Motion in One Dimension Ahmed H. Ali
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(10)
One Dimensional Motion with Variable Acceleration
The procedure is the same as we have done in the previous section. The velocity
vector is
(11)
Then the acceleration is given as
(12)
Since the acceleration is variable, then it can’t be written as a constant. We should
write the equation with respect to the velocity vector components, taking
derivation of them. So that,
(13)
In the one-dimensional system, we can write the equation;
Question 3.2: Assume that a car traveling at a constant speed of 30m/s passes a police car at rest.
The policeman starts to move at the moment the speeder passes his car and accelerates at a constant
rate of 3.0m/s2 until he pulls even with the speeding car.
a) Find the time required for the policeman to catch the speeder and
b) Find the distance traveled during the chase.
Solution 3.2: We are given, for the speeder:
vs of speeder= 30m/sec, constant speed, then
and for the policeman
a) The distance traveled by the speeder is given as . Distance traveled by
Mechanics Motion in One Dimension Ahmed H. Ali
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policeman
. When the policeman catches the speeder or
Solving for t we have t =0 and t=2/3 (30) =20s. The first solution tells us that the
speeder and the policeman started at the same point at t =0, and the second one
tells us that it takes 20 s for the policeman to catch up to the speeder.
b) Substituting back in above we find the distance that the speeder has taken
=0+0+1/2(3)(20)2=600m
H.w
Question 3.4:
A train with a constant speed of 60km/h goes east for 40min. Then it goes45 0 north-east for
20min. And finally it goes west for 50min. What is the average velocity of the train?