Vectors

Introduction to Vectors:

"The pendulum of the mind alternates between sense and nonsense, not between right and wrong."

    A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity's magnitude. Although a vector has magnitude and direction, it does not have a position. That is, as long as it's length is not changed, a vector is not altered if it is displaced parallel to itself.

Applications on Everyday Life:

  Aircraft in flight are subject to the direction of the winds in which the aircraft is operating. For example, an aircraft in flight that is pointed directly north along its longitudinal axis will, generally, fly in that northerly direction. However, if there is a west wind in the air in which the aircraft is flying, then the actual trajectory of the aircraft will be slightly to the east of north. If the aircraft was landing on a northbound runway, it would need to compensate for this easterly component of velocity caused by the west crosswind.

  We experience and interpret vectors in our everyday lives whether through photos, diagrams, architecture, or advertisements. Vectors tell us visually what we would normally say verbally. 


Vector Operations:

Graphical Vector Addition

 

  Adding two vectors A and B graphically can be visualized like two successive walks, with the vector sum being the vector distance from the beginning to the end point. Representing the vectors by arrows drawn to scale, the beginning of vector B is placed at the end of vector A. The vector sum R can be drawn as the vector from the beginning to the end point. The process can be done mathematically by finding the components of A and B,combining to form the components of R.

Components of a Vector:

Finding the components of vectors for vector addition involves forming a right triangle from each vector and using the standard triangle trigonometry.

1.)    

  

2.)

3.)

sources: [http://www.britannica.com/EBchecked/topic/1240588/vector]

                [http://www.physicsclassroom.com/class/vectors/u3l1a.cfm]

                [http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html]

               

                                              

Free Fall, One Dimensional Motion with Constant Acceleration

Introduction to Free Fall:


     Living young, wild and free. :)

Free fall is any motion of a body where its weight is the only force acting upon it. These conditions produce an inertial trajectory so long as gravity remains the only force. Since this definition does not specify velocity, it also applies to objects initially moving upward. Since free fall in the absence of forces other than gravity produces weightlessness or "zero-g," sometimes any condition of weightlessness due to inertial motion is referred to as free-fall. This may also apply to weightlessness produced because the body is far from a gravitating body. (Wikipedia)

A free falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects:

• Free-falling objects do not encounter air resistance.
• All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s^2

Examples of objects in free fall include:

• An object dropped at the top of a drop tube.

• An object thrown upward.
• The moon orbiting around the Earth. (Any planet or satellite is in free fall as their motion is caused entirely by gravity.)

History of Free Fall

The teachings of the great ancient wise Aristotle stating that heavy objects fall faster than light ones were accepted until the XVI Century. We know that if we drop a hammer and a feather or a sheet of paper from the same height, the hammer will reach first the ground. If we crumple the paper giving it a ball shape it is observed that both objects will reach the ground almost at the same time. It was the famous Italian physicist Galileo Galilei who refuted Aristotle's idea stating that, in absence of air resistance all objects fall with the same uniform acceleration. He cleverly proved his hypothesis using inclined planes getting a slower movement which could be measured with the rudimentary watches of that age. The slope of the planes could be increased gradually until reaching almost a vertical position allowing him to predict behavior of objects in free fall.

Acceleration of Free Fall

The magnitude of this free fall acceleration is denoted by the symbol g, whose value slightly varies with the altitude and the latitude. The symbol g is also called as acceleration due to gravity. Near the surface of the Earth the g value is 9.8m/s^2. The free fall is a known example of uniformly accelerated movement, with an acceleration, a = -g = -9.8m/s² (vertical axis pointing vertically upward). If you choose the vertical axis pointing vertically downward, the acceleration is taken as + 9.8m/s². The kinematic equations for a rectilinear movement under the acceleration of gravity are the same as any movement with constant acceleration:

 where   V_{f= final velocity}     V_i= initial velocity     
            t = time                     a= -g = acceleration due to gravity    
           Δy =displacement along the y-axis 


sources: [http://www.jfinternational.com/ph/free-fall.html]
               [http://www.physicsclassroom.com/class/1dkin/u1l5a.cfm]