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 movesThe 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 projectileIt 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

To find the maximum height h we use the equation

by substituting for the time t1 in the above equation

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?

SolutionConsider 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 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 ”.

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 F1 can be broken into two forces: Fx and Fy.

Section 2. Two Dimensional Forces into OneYou 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]

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.

Section 3: One Dimensional Forces into TwoYou 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 beFFred = cos 45 * 100 = 70.7 N

and the force by Jack isFJack = 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 DirectionsIf 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/Mechanics/Motion_in_Two_Dimensions http://www.staff.amu.edu.pl/~romangoc/M2-motion-two-three-dimensions.html http://library.thinkquest.org/10796/ch5/ch5.htm#Sec0 http://hazemsakeek.com/Physics_Lectures/Mechanics/includegp1lectuers_5.htm