Movement Author: Silvia Sokolovsky
Introduction:
"Imagine a perfect mystery. This kind of story has all the information and vital clues and prompts us to decipher the mystery of our own," so begins the Albert Einstein book The Evolution of Physics, and is valid to introduce the topic. Although your interest is very far from urging the genius of the twentieth century to solve the mystery that is contained within the problems will have to play detective and find the data available, make them understandable and consistent through reasoning. Which at first glance not so easy.
First, we introduce the problem of the movement, its causes and effects.
"Our intuitive concept of movement was linked to acts of push, lift, drag ... ... It seems natural to infer (conclude) that the greater the action exerted on a body, the greater its speed ... (Imagine pushing a car, if you push two people will faster than if you push it) ... The method of reasoning given by intuition was wrong and led to misconceptions regarding the movement of bodies. "
Suppose you want to skate on the floor, obviously will travel some distance and then we stop. If we go further we grease or oil the wheel axles of our skates and smooth as possible the way. What are we really doing? We are reducing the friction with the ground, friction.
Theoretically, if we imagine a path perfectly flat and skates with wheels without any friction, there would be any cause to oppose our movement, would be eternal.
see clearly that failure to push or drag a body, or is applied an external force, it moves uniformly, ie with constant speed in a straight line.
"This conclusion was reached by imagining an idealized experiment that can never be verified, since it is impossible to eliminate all outside influence" Einstein was primarily a theoretical physicist, as he imagined the experiences and applying known physical laws and mathematical elements trying to solve problems he himself raised. In your case, the problems will be given by the teacher, but if Einstein was served his "technical" Why do not you? ...
In the words of Einstein: "All movements observed in nature - for example, the fall of a stone in the air, a ship sailing the sea, a car moving down the street - are actually very intricate (difficult to understand.) To understand these phenomena is advisable to start with simple examples and gradually move to more complicated cases. "Let's case.
Movement: How
we realize that we are moving?.
not touch the mouse (mouse) for your computer while watching the second hand on your watch. As time passes, the mouse does not change position, but the second if. The mouse is stationary and the second hand is moving. Simply put, we realize that "something" moves to see how it changes its position as time passes.
The movement is the position change with time.
Suppose we have a stopwatch to "that time", every moment we designate by letter, usually often used the letter t. The moment we begin to measure is the time zero, so we can appoint him to (I sub-zero), and also may be indicated in the subscript the moment that is mobile. For example, if 5 seconds pass as we indicate t5. If we take two seconds
any, difference between the two will show the elapsed time between two moments: Dt = t - ti (subscript i indicates that the initial time interval).
This symbol D (differential) is a mathematical element that is used to indicate subtraction, "difference" between two values \u200b\u200bof a variable.
If the movement is horizontal to the ground can be considered as if the x-axis (x axis), so that every position is designated by the letter x. The position corresponding to time zero (to) is designated, then, as xo. The difference between any two positions allows us to calculate the space between them: Dx = x - xi
uniform linear motion (MRU )
easier movement is movement in a straight line (logically called straight) As any movement can be described by the space that is covered in unit time, suppose we go always the same amount of space per unit time. Imagine that for every second we walk two meters. In the first second we walk two meters, the second will have done four, the third six and so on ...
To further facilitate our study imagine that we start from the zero position at time zero. Locate our assumption on a table:
Instant (t)
0
1 2 3
4 5 6
7 8 9
10
Position (x)
0
2
4 6 8
10
12
14 16 18 20
Space and time are directly proportional mathematically, this means that if we divide each position for the moment is give us a constant value.
Physically this constant value, the ratio of the distance traveled and time elapsed, it is called speed.
So the speed in this type of movement is constant, as shown in the graph of speed versus time (v (t)) where speed is represented. If we chart the position at every moment which is indicated in the table, we see that we find a line. If you look closely the picture we can see that position at every moment can be calculated by multiplying the time (t) by the velocity (v), thus we have: x = v. Dt
does not have to start from scratch, so the positions can be determined by adding the position where we started, position (x), and what goes (Dt.v).
Suppose we start from the position 2, x = 2 m, and the speed is 2m/seg. let us add 2 m above position:
Instant (t)
0
1 2 3
4 5 6
7 8 9
10
Position (x)
2
4 6 8
10
12 16
14 18 20
22
is interesting that we get a straight line whose slope is the speed (2) and the intercept is the starting position (2): mathematical equation obtained is: x = 2Dt + 2. (I use the variables listed in the chart). Thus
space equation as a function of time from now we will call time equation, we write: x = x + v. Dt
vector and scalar Magnitudes: The numbers are abstract entities that by themselves mean nothing. That is its greatest virtue, then we assign the meaning we want. A simple three, as the occasion may be a sum of money, a bad note, whatever ... Everything we can measure can be represented by a number. Anything measurable will be called, then magnitude. And the magnitudes can be divided into two groups: scalar and vector. Suppose we are looking
cars moving on a straight street, all cars have the same direction (the street) but do not have to go to one side, may have a different sense. Movement is important in indicating the direction (line to which it belongs) and the direction it is moving. In mathematics there is an element that indicates meaning and direction in addition to the module (amount of speed) is the vector. For every variable that can be represented by a vector we will call "vector quantity." What
indicates the vector logic is used to indicate the speed of a car. Velocity is a vector quantity and its magnitude indicates the scalar part, the amount represents. Vector is indicated by enclosing the two lines: v. The module is always a positive value.
course are magnitudes that can not be represented by a vector, ie the time. The variables that we can only indicate their amounts are called scalar quantities. To better understand their differences explain a typical example:
difference between distance traveled and displacement: We were talking about position (x), space (Dx) and, although not named, travel. But these three words have different meanings in physics. Suppose you're in a corner, this will be your starting position and to make things from there begin to have so x = 0 m. Now walk two blocks on the same block. The distance traveled is 200 m, since each block is 100 m, but the displacement, the straight line joining the two positions, if we apply Pythagoras (see figure) will be 141.42 m. Moreover, if you turn apple, space travel has to be 400 m. but the zero displacement.
Displacement is a vector, the space covered a scalar, just a number. Uniform linear motion
Miscellaneous ( MRUV )
a little closer to the movement in the real world, we see that the speed is not the same throughout the journey. While the module changes, does not change in any way, but that depends on a third variable, the acceleration.
Acceleration:
Imagine you are traveling with a velocity v and doubled.
Its variation is: Dv = 2v - v = v (1). This change brings a certain time.
Now suppose that the rate tripled, the change is: Dv = 3v - v = 2v (2)
If we compare (1) and (2) we see that the variation in speed has been doubled. What happened to the time interval?. Obviously we need more time, exactly twice.
recap, the variation in the speed doubles and the interval of time required increases proportionately. The explanation is that there is a relationship between two variables are directly proportional. Therefore if we divide we get a constant, the aspect ratio between the acceleration.
must be noted that the relationship is between the speed variation and the interval of time NOT to do with speed. As the speed
a vector quantity and a scalar time, any mathematical operation between them will result in a vector, we can therefore deduce that the acceleration is also a vector acceleration units
: Applying the definition of acceleration, velocity variation with time, we will analyze your drives.
We can measure the speed in m / sec, so take the time unit in seconds to mathematically operate smoothly.
also be expressed as. Getting
Primitive function: To find the equations of motion (primitive function, mathematically speaking) can proceed by comprehensive or obtaining the area under the curve. As many of you can ignore them mechanisms of mathematical analysis, we use the second option.
MRUV The speed varies, but not in any way, depends on the acceleration and this is constant. If we look closely at the graph of acceleration versus time (graphic acceleration) we can realize that, no matter the chosen moment, "a" will always have the same value.
Suppose that the acceleration is 2 m/s2 when we start from the position 1 m. with a speed of 1 m / s
remember: x = 1 m v = 1 m / s.
If you look closely at the area which is determined from the graph of acceleration and the time axis, indicated by the successive time intervals from zero (dotted lines) we see three figures, ie three rectangles.
first interval [0, 1]
Second Interval [1, 2]
Third Range [2, 3]
Area = base. Height
In a rectangle, any side can be base or height. To facilitate further calculations we will take the time interval (t) as high based Þ = a, height = Dt Area = a. Dt
The acceleration determines how velocity and the area under its graph indicates the rate at the end of this time interval: Area = v, thus we have: v = a. Dt
not forget that at the beginning of this movement speed was not void Þ v = v + a. Dt (Equation 1) (This equation allows us to calculate the velocity at every instant, or instant speed.)
Complete the following table based on the data using equation 1.
to
to Dt Dt Dt + v
to
v
2
0
2. 0 = 0
2. 0 + 1 = 0 + 1 = 1 2
1 1
2. 1 = 2
2. 1 + 1 = 2 + 1 = 3
2 2 3
2. 2 = 4
2. 2 + 1 = 4 + 1 = 5 5
2 3
2. 3 = 6
2. 3 + 1 = 6 + 1 = 7 7
Take the points whose coordinates are determined by (t, vt) (columns in color) and delivered to each of the speed chart . We see that the speed varies depending on the time gives us a line.
Whenever a variable dependent on a constant give a line on the graph.
Again take the time intervals [0.1] [0.2] and [0.3]. Below the line are determined three trapezoids.
new coach will be the height, the bases (the trapezoid has two) will be the speed. The vo (initial velocity) is the smaller base while vt (instantaneous velocity) is the largest base.
We have already seen that the rate indicates how much space is traversed per unit time, thus varying the speed changes the amount of space covered by each time interval equal duration. and the area under the vt graph indicates the position of the body at the end of time slot. Given that we start from the position 1 m. (X = 1 m) we have:
v v + vo vo
(v + v): 2
Dt [(v + vo) 2].
Dt [(v + vo) 2]. Xt dt + xo
1
1 1 + 1 = 2
2: 2 = 1
0
1. 0 = 0
0 + 1 = 3
1 1
1 + 3 = 4
4: 2 = 2
1
2. 1 = 2
2 + 1 = 3
5
1 1 + 5 = 6
6: 2 = 3 2
3. 2 = 6 6 + 1 =
7 7
1
1 + 7 = 8
8: 2 = 4 3
4. 3 = 12 = 12 + 1
Take 13 points (Dt, x) (columns in color). Putting them into graphic space depending on the time we obtained a curve, a parabola.
Whenever a variable depends on another variable we get a curve as a graph.
(not the equation is commonly used to find xt, we replace vt by equation 1), we as (mathematical operations)
This equation, called time equation is the most frequently used to find xt. From Equation 1 and Equation 2, math that are on your own, get a third equation to provide enough resolution of problems: 2. Dx. a = v 2 - v 2 (Equation 3)
Using equations 1, 2 and 3 can solve any problem
MRUV Freefall:'s assume that we are on top of a bridge 30 meters high watching the water pass. For fun, we drop a stone and fall time measured with a stopwatch. Every time I let go of each stone will trace a straight path from our fingers into the water. No matter how many times we do this simple experiment, always fall in the same way. Obviously, the falling stone produces a linear motion. Now we must ask
what happens with speed. As we release the stone can safely be assumed that initial velocity is zero. When the initial velocity is zero, the body starts from rest. Certainly the speed of the stone does not remain constant, otherwise it would float if we drop. So it is ruled that the motion of falling even (MRU). The speed changes, we realize intuitively that accelerates. With all these data we can assume that the fall of any object is accelerated rectilinear motion (MRUV).
no longer use the name "x" for the various positions taken by the body along its path, but being a vertical movement use a "y". The initial position (the height at which we let the stone) I will be designated as the initial instant of motion our stopwatch should be zero. that way the space covered by the falling body (30 meters) will be designated as Dy (Dy = 30 m).
acceleration of gravity: It is interesting that every time the stone falls, taking time with our clock, it takes 2.47 seconds to touch the water surface. To verify that the observed effect is not the type of item that you drop, take a role and make him a bun (very tight) and leave it down. Also take 2.47 seconds to fall. How is it possible?!. Simply, as stated, the free fall path is straight, rectilinear motion and speed change suffered by both bodies is the same. Both the stone and paper, thrown with the same initial speed and from the same height, fall through accelerated rectilinear motion. Let
calculations to determine the value of the acceleration with falling:
Replace the value of each piece of information: vo = 0 m / sec., Dt = 2.47 sec. and Dy = 30 m.
*
Regardless of body mass or the height at which fall, any object left in free fall experience the same acceleration that from now on we will call the acceleration of gravity and is designated with the letter g.
The acceleration of gravity, as any acceleration is a vector. The direction of this vector is vertical, and the fact that a body falling, it will accelerate, indicates that the direction of the acceleration vector of gravities "down."
The acceleration of gravity is the same for any body, no matter its mass, from the same height and with the same initial velocity, if we drop a needle, a bucket of sand or a plane, the three will fall at the same time and arrive at the same speed. Nothing better than experience to make sure that the variation of speed and downtime, not dependent on body weight but the acceleration of gravity (G). Count the time it takes to drop several objects (rubber, pencil, etc) and draw your own conclusions ... Vertical Shooting
: When throwing a stone up, we have two possibilities: that the path is straight or not. The second case we will to get the motion in two dimensions, while reasoning that occurs when pulling "vertically" a stone upwards. First
analyze whether the vertical shaft is accelerated or slowed movement.
The speed with which we throw a stone vertically upward, initial velocity, it must be different from zero, but would fall. The body goes up until it stops at a position which we call the maximum height (Ymax). In this position, in which the object is stopped, the speed must be zero. We are facing a movement slowed.
For convenience, let's put on the meaning of the initial velocity plus sign. To put it easier, initial speed is always positive, hence its effect will be positive. Any vector that has the same sense that the velocity is positive and one that goes in the opposite direction will be negative.
This movement is decelerated, the velocity and acceleration have different meaning, signs are opposite, then we conclude that gravity has a negative sign. g = - 9.8 m/s2. *
Importantly, when the stone reaches its maximum height and begin to fall, the sign of its velocity (during the fall) will also be negative.
Thus, for vertical shooting and free-fall can be used: as a time equation.
* In problems, to make it easier to order, we will use as the value of gravity "- 10 m/s2.
How do you solve a problem?
To solve a problem you always have to follow three steps:
1. Search the data of the problem and which serve to distinguish those who do not.
2. Find the unknown, we can not solve any problem if we are not quite clear what is being sought.
3. Enforce laws and equations that are consistent with data collected.
Example of how to solve a problem: F
guy drops stones from the balcony of his house. The goalkeeper, who is on the sidewalk, you notice that one of the stones takes 0.2 sec. in passing by the front door, which is 2m high. With this information, found above the floor to leave the stones. (Hint: take a reference system with the origin at the upper edge of the door).
The fact that the door is 2 m (Dy), the acceleration of gravity (which the stone falls to the floor can be positive as the speed and severity of the body have the same meaning), with an interval of time 0.2 sec. (Dt). We can apply the equation to calculate time speed has to get to the top of the door (v1)
To calculate the height of the building (from the door to the terrace) we use the initial velocity (as the leaves fall from rest), the speed just to find and gravity.
2. g. Dy = v2 - vo2 ® Dy
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