Download motion in two dimension

Motion in Two
Dimensions
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.
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
Since the acceleration is constant then we can
substitute
this give
Then
Since our particle moves in two dimension x and y with
constant acceleration then
but
●
ballistics considers as projectile motion
Example
A good example of the motion in two dimension it the motion of
projectile. To analyze this motion lets assume that at time t=0
the projectile start at the point xo=yo=0 with initial velocity vo
which makes an angle qo, as shown in Figure 2.5.
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 Vy=0
by substituting in equation
we use the equation h To find the maximum height
in the above equation 1t by substituting for the time
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?
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
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.
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
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
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.
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 .
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
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 .
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.
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
It was already .much smaller then the velocity of light
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
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.
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 .
Let’s consider an example of motion in two dimensions – it is easier to
make the drawing for a such case.
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
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 ,vyC .0=
is moving along y axis, so the x component of its velocity is
zero ,vxD .0=
Section 1 :Simple Breakdown of Forces
You can break down forces into several components easily. For example, the
force F 1can be broken into two forces: Fx and Fy.
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
30N + (cos 45 * 40) N = 58.3 N. [E ]
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
0N + (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.
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
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 .)
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 .
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).
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.
Sources :
http://physics.bu.edu/~duffy/py105.html
http://en.wikibooks.org/wiki/Physics_with_Calculus/Mechan
ics/Motion_in_Two_Dimensions
http://www.staff.amu.edu.pl/~romangoc/M2-motion-twothree-dimensions.html
http://library.thinkquest.org/10796/ch5/ch5.htm#Sec0
http://hazemsakeek.com/Physics_Lectures/Mechanics/incl
udegp1lectuers_5.htm