Vectors

A vector is formally defined as an element of a vector space.

A zero vector, denoted , is a vector of length 0, and thus has all components equal to zero.

Two Fundamental Definitions:

•  Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points.

              

•        A vector having the same magnitude as A but in the opposite direction to A is denoted by -A

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.

SAMPLE PROBLEMS

1. You travel for 3.0 m/s in an angle of 45 deg (North of East) and then proceeded to 5.0 m/s, 135 deg (North of West). How far from where you started did you end up? Draw the vector and indicate the scale you used.

Therefore, your resultant vector would be 5.83 m/s, 104 deg (North of West).

2. You are jogging at the speed of 4.0 m/s, 135 deg (North of West)  and then turned south of east at 4.0 m/s, 315 deg. How fast from where you started did you end up? Draw the vector and indicate the scale used.

Therefore, your resultant vector would be 0 m/s since they have different directions and the same magnitude. 

3. I am riding a bike at 7.0 m/s at the east and then going to the north at 2.0 m/s. How fast from where I started did I end up? Draw the vector and indicate the scale used. 

I traveled for 7.28 m/s at an angle of 15.9 deg, North of East.

4. My sister walked at 4.0 m/s, North and turned East at  2.0 m/s. She then walked farther at 2.0 m/s at an angle of 210 deg (South of West). How fast from where she started did she end up?

She walked at 3.01 m/s at an angle of 84.9 deg, North of East.

Free Fall

I'm falling and it feels so free!

Free fall. Every time I hear this word, that picture on the left pops into my head. Falling freely neglecting air resistance.

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/s

Because free-falling objects are accelerating downwards at a rate of 9.8 m/s/s, a dot diagram of its motion would depict an acceleration.  If an object travels downward and speeds up, then its acceleration is downward.

Free-fall acceleration is often witnessed in a physics classroom by means of an ever-popular strobe light demonstration. The room is darkened and a jug full of water is connected by a tube to a medicine dropper. The dropper drips water and the strobe illuminates the falling droplets at a regular rate - say once every 0.2 seconds. Instead of seeing a stream of water free-falling from the medicine dropper, several consecutive drops with increasing separation distance are seen.

A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the acceleration of gravity - the acceleration for any object moving under the sole influence of gravity. A matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it - the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s. 

A position versus time graph for a free-falling object is shown below.

Observe that the line on the graph curves. A curved line on a position versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s/s), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object, the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity.

A velocity versus time graph for a free-falling object is shown below. 

Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s/s, downward), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object that is moving in the negative direction and speeding up is said to have a negative acceleration. Since the slope of any velocity versus time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 9.8 m/s/s in the downward direction.

(Source: http://www.physicsclassroom.com/class/1dkin/u1l5a.cfm)

And that’s what I’ve learned so far about free fall. Thanks for checking this page out! :)