Motion in Two Dimensions

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Motion in Two Dimensions
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Motion in two dimensions like the motion of
projectiles and satellites and the motion of
charged particles in electric fields.  Here we
shall treat the motion in plane with constant
acceleration and uniform circular motion.
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Motion in two dimension with constant
acceleration Assume that the magnitude and
direction of the acceleration remain unchanged
during the motion.  The position vector for a
particle moving in two dimensions (xy plane) can
be written as                where x, y, and
r change with time as the particle moves The
velocity of the particle is given by  
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Since the acceleration is constant then we can
substitute this give             Then
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Since our particle moves in two dimension x and y
with constant acceleration then
  but    ? ballistics considers as
projectile motion
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Example A good example of the motion in two
dimension it the motion of projectile.  To
analyze this motion lets assume that at time t0
the projectile start at the point xoyo0 with
initial velocity vo which makes an angle qo, as
shown in Figure 2.5.                
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Horizontal range and maximum height of a
projectile It is very important to work out the
range (R) and the maximum height (h) of the
projectile motion.
To find the maximum height h we use the fact that
at the maximum height the vertical velocity Vy0
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by substituting in equation
To find the maximum height h we use the equation
by substituting for the time t1 in the above
equation
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Example                                      
                                                  
                                                  
                    Suppose that in the
example above the object had been thrown upward
at an angle of 37o to the horizontal with a
velocity of 10m/s.  Where would it land?
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Solution Consider the vertical motion To find
the time of flight we can use since we take
the top of the building is the origin the we
substitute for   Consider the horizontal
motion then the value of x is given
by            
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Motion in More Than One Dimension It is an
intriguing result that the motion of a particle
in one direction does not affect the motion in
any perpendicular direction. The classic example
is if you shoot a gun level to the ground and
drop a bullet at the same time, they hit the
ground at the same if they started at the same
height. That is, the motion of the bullet
horizontally has absolutely no affect on how it
moves vertically. You might ask why is it then
that a paper airplane thrown and dropped do not
hit the ground at the same time -- the answer is
that the situation is fundamentally different
because you have interaction with the air. A
mathematically precise way of saying this is that
the velocity really is a vector. It adds like a
vector, and you can split it up into components
like a vector.
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Using this notion, let's derive the constant
acceleration equations for vectors and you can
see how vectors are very useful.
where the subscripts indicate the ith component
of the vector.Integrating each component (or
equivalently, integrating the vectors
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It's easy to see that everything is working out
in exactly the same way it did for the 1
dimensional case, except we're doing it in every
component! Thus we have
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and with some messy vector algebra,
The last formula, as you can probably guess, is
completely useless. But that's what it is if you
were wondering. A hat above a vector is the
vector divided by its magnitude, making it a unit
vector in the direction of the original vector.
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Motion in Two and Three Dimensions
This is a more common type of motion compared to
the very restricted class of motions along a
straight line. If you drive a car you turn left,
right, and you go uphill and downhill even if
this up and down motion is difficult to notice.
When traveling by air three dimensions of motion
are even more clearly pronounced. For
simplicity of representing the motion on graphs
we will mostly discuss examples of two
dimensional motion, but remember, adding movement
in a third dimension does not change the basis of
the theoretical description presented in this
tutorial. Whenever we talk about motion in two
dimensions the statements are also true for
motion in three dimensions and vice versa. 
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There is one fundamental law concerning motion in
two or three dimensions it can be decomposed
into three independent motions each along one
direction. A very classic example of this
statement will be described later it is
projectile motion. Now consider a quite common
example a person crossing a river in a boat.
This motion is schematically shown
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The boat starts perpendicularly to the bank line
and so is the direction of the line drawn along
the boat. The person on the boat has the feeling
that he travels perpendicularly to the river
bank, but an observer on solid ground sees the
boat moving along a line that is tilted away from
the perpendicular one. The truth is that boat is
moving independently and simultaneously into two
directions perpendicularly to the bank line
the green arrow indicates its velocity in this
direction and parallel to the bank line, with
the river stream as indicated by a dark blue
arrow representing the velocity in this
direction.  
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The resulting velocity is indicated by the light
blue arrow along the red arrow indicating the
resulting direction of motion. The two velocities
green and blue are independent. The velocity
of the river current does not depend on the
velocity of boat that is obvious. The velocity
of the boat, relative to the water,  does not
depend on the velocity of the river current. We
are not talking about mountain rivers, where the
current may be so fast and turbulent that it
prevents the motion of a boat powered by a small
engine. You can imagine moving very slowly across
a very wide river, so that before reaching the
opposite bank the boat will have moved downwards
a few centimeters. This will be a three
dimensional motion and motion in the third
direction is also independent from the motions in
the two other directions. You can find many
other examples of motions of an object with the
resulting displacement being the sum of
displacements in two or tree dimensions.
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Motion of an object in one direction is
independent of motion of the same object in
another direction or directions.   This is true
if we do not consider the influence of forces on
motion, that is if we study Kinematics, and for
velocities much smaller then the velocity of
light. It was already mentioned that velocity or
speed of the order of 1000 km/s (kilometers per
second!) may be considered as negligible compared
to the speed of light, therefore all examples of
motion we will describe in this part of the
tutorial fulfill this criteria of much smaller
then the velocity of light.
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The independence of motion in different
directions is the basis for analysis of all
examples of motion in this paragraph. The
independence we are talking about holds not only
for motions in different directions. Think about
traveling by train or tram. If you walk around
the carriage this motion is independent from the
motion of the wagon itself even if you move in
the same direction as a wagon. We exclude the
effect of the shaking of the carriage, which may
cause some difficulties in walking. It is easy
to imagine simultaneous motion of an object in
more than three directions if you combine the
motion of a person in the train with the motion
of the Earth about the axis and around the Sun.
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For experimental purposes four moving platforms
can be constructed, each one smaller then the
previous, each radio-controlled. They can be put
one on another and each moved in slightly
different directions. The platform on top will
experience simultaneous motion in four
directions. For a well leveled  platform though,
the motion will only be in two dimensions. Do not
confuse direction with dimension. There is
indefinite number of directions the object can
move along, but there are only three independent
dimensions in space. What does independent
dimension mean? In the Cartesian coordinate
system the directions of x, y and z axis are
independent. If the motion of an object is along
the x axis it is not possible to create such a
motion by any combinations of motions along the y
and z axis. The same is true for motion along any
of the axis it cannot be replaced by any
combination of motion along two other axis.
Lets consider an example of motion in two
dimensions it is easier to make the drawing for
a such case.
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The straight line motion in an arbitrary
direction (except the previously described motion
along the x or y axes) can be decomposed into
simultaneous motions in x and y directions. This
situation is depicted
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Motion of objects in different directions
decomposed into motions in two independent
directions x and y. These axes define a two
dimensional space (in common language simply a
plane). Small red circles represent moving
objects, red arrows their velocities. For objects
denoted A and B those red velocities can be
decomposed into two independent velocities along
x and y axes. Object C is moving parallel to the
x axis, therefore its velocity cannot be
decomposed into any other direction. Or, formally
you can say that y component of its velocity is
zero, vyC0. is moving along y axis, so the x
component of its velocity is zero, vxD0.
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Section 1 Simple Breakdown of Forces You can
break down forces into several components easily.
For example, the force F1 can be broken into two
forces Fx and Fy.
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Section 2. Two Dimensional Forces into One You
can combine two forces into one. Suppose Jack
pushed a box with a force of 30 N at 0 degree and
Michael pushed it with a force of 40 N at 45
degrees. How can we find the net force acting on
the box?
The first thing you have to do is to find all
forces on x direction (x axis) only. Jack exerts
30 N and Michael exerts (cos 45 40) N at x
direction. Therefore, the total force on x
direction would be
30 N (cos 45 40) N 58.3 N. E
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Then, you will have to analyze all forces on y
direction (y axis). Since Jack exerts no force
and Michael exerts (sin 45 40) N, the total
force on y direction would be 0 N (sin 45
40) N 28.3 N. N To find the combination of
Jack and Michael's forces, we can just combine
forces on x and y directions. Therefore, using
the Pythagorean Theorem, we can calculate that
N is the
magnitude (size) of the combined forces.
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Section 3 One Dimensional Forces into Two You
can also break down forces. For example, Fred
pushed a box to the east and Jack pushed it to
the north. If the net force is 100 N to north
east by 45 degrees,
the force applied by Fred would be FFred cos 45
100 70.7 N and the force by Jack is FJack
sin 45 100 70.7 N
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Section 4. Forces involving Gravity When you
place a box on an inclined plane, the box will
slide. What is the force that makes it slide?
First, the force of gravity is acting on the box.
The force of gravity acts perpendicular to the
horizontal ground. Also, the normal force is
acting on the box since it is on the inclined
plane. (The normal force acts on all objects on
the ground.) The normal force always acts
perpendicular to the surface, not to the
horizontal. If the plane has an incline of x
degrees, then
FN Fg cos x
since FN is leaning x degrees to the left (Fg is
the force of gravity).
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There is also a force of friction between the box
and the plane. It acts parallel to the surface,
not to the horizontal. The below drawing
summarizes the forces acting on the box
When you combine FN and Fg, a single force that
acts parallel to the surface will be generated.
This force, called the force of parallel (F//),
causes the box to move forward. F// can be
calculated by Fg sin x.
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To conclude, the mixture of the force of parallel
and the force of friction determines how the box
moves. If the force of parallel is larger than
the force of friction, the box will slide. If
both forces have equal magnitude, the box will
not slide. If the force of friction is larger
than the force of parallel, the box will move
upward. (Just kidding. The force of friction can
never be greater than the force of parallel).
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Section 5. Forces in Three Directions If you see
a mixture of three or more forces like below,
All you have to do is to calculate forces on x
direction, on y direction, and add these two
forces into one to get the total net force.
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Sources http//physics.bu.edu/duffy/py105.html
 http//en.wikibooks.org/wiki/Physics_with_Calcul
us/Mechanics/Motion_in_Two_Dimensions http//www
.staff.amu.edu.pl/romangoc/M2-motion-two-three-di
mensions.html http//library.thinkquest.org/1079
6/ch5/ch5.htmSec0 http//hazemsakeek.com/Physic
s_Lectures/Mechanics/includegp1lectuers_5.htm
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