Lesson 6 Newton's Laws

The most important law of physics is the equation F = ma. This simple yet complex equation is a vector equation that is the basis for mechanics, the science of motion. However there are two problems with this equation. In order for this equation to be true both F and a must have the same direction. Also, acceleration is a second derivative and does not provide position. Therefore F = -mg^z where (^z is downward position). This equation is not only for falling bodies or projectiles on which air resistance and gravity act, but for all motion. This equation is the fundamental basis for all of Newton's laws.

Newton's Laws of Motion:

1.) Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it. (Embraces inertia)

2.) The change in motion is proportional to the force impressed and is made in the direction of the straight line of which the force is impressed.

p = mv (momentum = motion)

F = dp
       dt
F = dmv
        dt
F = m dv
           dt
F = ma

3.) For every action there is an equal and opposite reaction.

These laws display how body's do not act but interact. This is because the laws did not change but did change how the world was perceived.

Newton's Laws are again explained in the following animation:





These laws also explain why objects projectiles follow a parabolic curve:

Horizontal - No force, constant speed

Fx = 0

Fx = max

max = 0

dvx = 0
 dt

vx = 0 (constant)
vx = vx0

Vertical - Force, constant acceleration

Fz = -mg

Fz = maz

maz = -mg
az = -g

dvz = -g
 dt

vz = -g

Because a projectile has both horizontal and vertical motion, these two statements above are simultaneous and explain the parabolic path that is followed by projectiles. This conclusion proved that the aristotelian idea that objects followed a straight path was false. However, as projectiles follow their parabolic path, the distance between the object and the path the object would follow if there was no gravity is equivalent to the square of time. In other words, the object's path coincides with Galileo's Law of Falling Bodies, specifically, the Law of Odd Numbers.

This is how the equation of a projectile is found:

dvz = -g

 dt

vz = -gt + c

vz = -gt + c

vz (t) = -gt + c
vz (0) = c

vz = -gt + vz0

dx = vx0
dt

x = vx0 + x0

dz  = -gt + vz0
dt

z = -1/2gt^2 + vz0 + z0

Below is a animation that displays the concept that it takes the same amount of time for a bullet to hit the ground after being dropped and fired:

Lesson 5 Vectors

Vectors are certain physical quantities with both direction and magnitude that serve as valuable tools for the United States Coast Guard. The Coast Gaurd uses the process of triangulation and must factor in wind speeds, current speeds, and many other vector represented values when searching for a lost or missing vessel. Below is an example of a vector and how one is indicated in a plane:

Displacement, velocity, and acceleration are all vectors and are represented by bolded symbols:

Displacement: s
Velocity: v
Acceleration: a

The magnitude of a vector is determined by values known as scalars. Time, mass, and magnitude are all scalars and are represented by italicized symbols:

Time: t
Mass: m
Magnitude: A or A

Vectors can be both added and subrtacted:

In the seventeenth century, Fermat and Descartes wrote the opening chapter on coordinate systems by connecting both geometry and algebra. However, all coordinate systems work equally well because the laws of physics work everywhere. And the vector expresses this fact.

The dot product of two vectors always produces a scalar. If the dot product is equal to zero, then the vectors are perpendicular to one another. The dot product is modeled in the image below:

From this, the angle between two vectors and the magnitude of a vector can be found:

Cos (theta) = A.B   gives the angle between two vectors.

                   [A] [B]

[A] = square root ((x^2) + (y^2)) gives the magnitude of the vector.



The cross product of two vectors A and B produces the vector AxB. The length or magnitude of AxB is equivalent to the area of the parallelogram formed by the original two vectors A and B:

Direction is determined by the right hand rule:





The right hand rule measures the tendencies of two vectors to be perpendicular. Vectors can also be used as effective tools for describing spinning objects.

There is a common notation used with vectors such that ^i and ^j are unit vectors <1,1>

^i x ^j = 0

^i x ^i = 1

^j x ^j = 1

<x^i,y^j>

These vectors can also be added, subtracted, multiplied, and divided by one another:

A = Ax^i + Ay^j

B = Bx^i + By^j

A+B = Ax^i + Ay^j + Bx^i + By^j

         = <Ax + Bx>^i + <Ay + By>^j

A . B = <Ax^i + Ay^j> . <Bx^i + By^j>

          = Ax^i . <Bx^i + By^j> + Ay^j . <Bx^i + By^j>

          = AxBx + AyBy

Vectors can also be represented in the three dimensional coordinate plane:

<x^i + y^j + z^k> is equal to the diagonal line drawn from the origin of i j and k in the image above.





Lesson 4 Inertia

Inertia encompasses the concept that any object moves at a constant speed in a straight line unless acted upon by an outside force. This concept was developed by Galileo Galilei. After his birth in 1564, Galileo developed the Law of Falling Bodies and experimented with inclined planes. Galileo did not cease his experimentation, but instead developed two telescopes, which were the source of his financial security throughout his life. Through these telescopes Galileo discovered the moons of Jupiter (Galilean satelites), the mountains on the surface of the moon, the phases of venus, and the black spots present on the sun. Through his studies of our solar system, Galileo confirmed that Earth, and the other planets in our solar system spin as they complete revolutions around the sun. Capurnicus's model of our solar system was in fact correct, however, this assertion did not explain why the inhabitants of the Earth do not feel as though they are moving as the Earth orbits around the sun.

Aristotle originally believed that all objects come to rest naturally but Galileo concluded, based upon the principle of inertia, that everything is in horizontal motion at the same speed and time as the Earth. Thus, humans do not feel as though they are in motion because there is no friction due to the force of gravity. From this enlightenment, Descartes developed the Law of Inertia and concluded that objects in motion remain in motion until something interferes. The universe is an example of an object that will never come to rest because absolute rest and motion do not exist.

Furthermore, whether an object is in motion or at rest completely depends on view point for all motion is relative.

In order to understand the principle of inertia, Galilieo said to picture a ball falling from the mast of a sailboat. If the sailboat is moving, the ball won't fall back into the water but will fall straight down on to the deck of the ship. On the other hand, if the ball was sitting at rest of the mast of another sailboat and then released while the other sailboat is in motion, the ball will fall back into the water. This is because objects fall in a parabolic curve. The principle of relative motion and inertia is displayed in the following video:



Without the classification of the idea of inertia, Newton would have never been able to compose his laws of mechanics.

Lesson 3 Derivatives

Kinematics is a branch of science that deals with motion in the abstract. In particular, particle kinematics deals with position, distance, velocity, speed, and acceleration. In order to understand these components of motion and their presence in the study of physics, one must explore derivatives. A derivative is said to be the "grammar of Differential Calculus" and details three major items. First, a derivative provides the slope of a line tangent to a curve at any point on the curve, Second, a derivative provides the velocity of an object at any time, and Third, a derivative provides the rate of change of a variable with respect to another variable. Slope, a major factor of Differentiation, is also a derivative:

Positive and Negative Slopes

Slope of a chord = Change in elevation                    (On a curve)

                        Change in horizontal distance

Slope of tangent line = derivative

Therefore, slope is the derivative of elevation with respect to horizontal distance.
Slope of tangent = dy
                               dx

There are also a number of rules that govern the ways in which a function is derived:

The Sum Rule

d (mx+b) = m

dx

d (y+z) = dy + dz

dx            dx    dx

The Product Rule

A= l x w

(Delta)A= (Delta)l x w

(Delta)A = l x (Delta)w

d (yz) = ydz + zdy

dx            dx      dx

d (xx) = x d x + x d x

dx              dx        dx

d x^2 = 2x d x

dx              dx

d (mx + b) = m

dx

d (1x + b) = 1

dx

d x = 1

dx

d x^2 = 2x

dx

d x^n = nx^n-1

dx

The Chain Rule

dy = dy dx

dt     dx dt

All of these rules contribute to the ease of mathematics when exploring complex theories. For example, Albert Einstein's Theory of Relativity is the most complex aspect of physics, and Einstein stated that he longed to use the simplicity of mathematics to support his theory. However, his theory was so elaborate that he could not rely on mathematics eniterly, but rather, use it as a tool because mathematics is limited in some scenarios. For instance, it is impossible to take the derivative of an absolute value function when x = 0. From Kinematics to calculus, mathematics and Differentiation are undeniably important.

*(Watch the following video for some in-depth studies of derivatives)*

Throughout this video you will view the concept of a derivative through the study of slope, elevation, and horizontal distance. The "Derivative Machine" will display the derivatives of trigonometric functions. The video concludes with the relationship between distance, velocity, and acceleration. The three graphs portray relationships between the graph of a function and its derivative. In fact, the derivative of a function is negative (below x-axis) when the function decreases and the derivative is positive (above x-axis) when the function increases.

 

Lesson 2 The Law of Falling Bodies

Galileo Galilei proposed that all bodies fall with the same constant acceleration because the effect of gravity is the same on all objects. However, on Earth, air resistence affects the rate at which objects fall. This is why a feather falls to the ground at a slower rate than a hammer on Earth, but in a vacuum, a volume of space that is empty of matter, both the hammer and feather would hit the ground at the same time. Galileo's theory was tested on the Apollo 15 mission, part of the American Apollo space program by David R. Scott. Scott held both a hammer and feather at the same distance from the ground and released them at the same time. The moon served as a perfect vacuum and the hammer and feather in fact landed on the surface of the moon at the same time. Galileo was correct!

Galileo not only proposed that bodies fall with the same constant acceleration but that their distance fallen is proportional to the odd numbers. This theory was known as the Law of Odd Numbers. This theory is displayed in the image below:

After each time period the object falls in an odd numbered interval. Additionally, the total distance fallen after each interval is equal to the perfect square. In other words, "distance fallen is proportional to the square of time." From this we can form the equation:

S(t) = ct^(2)

Where (c) represents a constant of about sixteen feet and (t) represents time, this equation can be used to find the distance traveled in a particular amount of time. For example, if a woman is on an amusement park ride and is free falling, what is her distance fallen after two seconds?:

S(2) = c(2)^(2)

S(2) = 4c

S(2) = 4(16)

S(2) = 64 ft.

The woman falls sixty-four feet after two seconds, but what is her speed after two seconds? The woman's speed can be calculated by using the equation:

S = d/t

Where (d) represents distance and (t) represents time.

S= 64ft./2sec.

S=32ft./sec. 

We have just found the woman's instantaneous speed, but what if we want to find her average speed as she is free falling on the ride from time (t) to time (h) seconds later?

Average Speed = Change is Distance

                            Change in Time

S(t) = ct^2

S(t+h) = c(t+h)^2

Average Speed = S(t+h) - S(t)

                             h

                           = c(t+h)^2-ct^2

                             h

(t+h)^2 = t^2+2th+h^2

c(t+h)^2 = ct^2+2cth+ch^2

                                      = ct^2+2cth+ch^2-ct^2

                                         h

                     = 2cth+ch^2

                        h

Average Speed = 2ct+ch

Instantaneous Speed = 2ct, therefore V(t) = 2ct (V represents Velocity)

Distance = S(t) = 16t^2 and V(t) = 32t

Thus speed (velocity) is the derivative of distance.

Now we must find the acceleration of the woman, which according to Galileo should be constant.

V(c) = 2ct

V(t+h) = 2c(t+h)

            = 2ct+2ch

Average Acceleration = 2ct+2ch-2ct

                                       h

                     = 2c

                                 A(t) = 2c (Always the same)

This mathematically proves that Galileo's theory was correct. Since A(t) = 2c and the force of gravity is always constant:

The force of Gravity (g) is equal to 2c

g = 2c

c = 1/2g

Distance: S(t) = 1/2gt^2

Speed (Velocity): V(t) = gt

Acceleration: A(t) = g 

As displayed above, derivatives are a vital component of physics. Ultimately accelerated motion is uniform by the odd numbered law proposed by Galileo.