CA 5.12 Acceleration Transformation

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COSMIC ADVENTURE 5.12 ACCELERATION TRANSFORMATION

ACCELERATION IN CLASSICAL PHYSICS

Cosmic Adventure 5.12a

Acceleration

When a rocket moves from zero velocity (v=0), that is at rest, to a high orbital speed (v), it is said to be under acceleration.

Acceleration & Deceleration

Like velocity, acceleration can be negative or positive. When positive, we call it acceleration; when negative, it is deceleration or retardation. For the convenience of our discussion, we refer to both cases as acceleration.

Mathematical Expression

In classical physics acceleration, usually symbolized by 𝑎, is defined as the change in velocity.

𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 [𝑎]

=𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒

𝑎 [𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛]

=∆𝑣 [𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦]

∆𝑡 [𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒]

=∆𝑣

∆𝑡

Vector Nature of Acceleration

Since acceleration a has a magnitude and direction, it is inherently a vector quantity. The operation of subtracting the initial from the final velocity, i.e. 𝑣𝑡 − 𝑣𝑜, is to be done by vector protocol. However, since we are dealing with objects moving in one direction, we do not have to be too concerned with the mathematics of vectors.

Average Rate of Velocity Change

The average acceleration is defined by:

𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 =𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒

𝒂𝑎𝑣𝑒𝑟𝑎𝑔𝑒

=𝑣𝑡 − 𝑣𝑜

𝑡𝑡 − 𝑡0=

𝛥𝒗

𝛥𝑡

Particle in motion With velocity 𝑣 at 𝑡𝑡

particle with velocity 𝑣𝑜 at 𝑡0

Instantaneous Rate of v Change

The instantaneous acceleration at any time can be obtained by taking the limit of the average acceleration as the time interval approaches zero:

𝒂𝒊𝒏𝒔𝒕𝒂𝒏𝒕𝒆𝒏𝒆𝒐𝒖𝒔 = lim𝒕→𝟎

𝛥𝒗

𝛥𝑡=

𝑑𝑣

𝑑𝑡=

𝑑2𝑧

𝑑𝑡2

In uniform motion a = dv/dt = 0.

Geometric Representation of Acceleration

In acceleration, the velocity changes from time to time. The graph of acceleration is different from that of velocity.

Δ𝑥

Δ𝑡 𝑣 =

Δ𝑥

Δ𝑡

Acceleration Velocity

Units of Acceleration

In the MKS system, acceleration is measured in meters per second in one second, or (𝑚/𝑠)/𝑠= 𝑚𝑠−2.

In CGS it is centimetres per second in one second, or (𝑐𝑚/𝑠)/𝑠 = 𝑐𝑚 · 𝑠−2.

In the Imperial system, in feet per second, or 𝑓𝑡/𝑠−2.

Example: Acceleration of

40 kilometres in one second or

30 miles per second

ACCELERATION IN RELATIVITY

Cosmic Adventure 5.12.b

Special Relativity & Inertial Frames

The theory of special relativity was initially meant to apply to inertial frames which are at rest or moving at constant speed. Accelerating frames are different to inertial frames. So it is thought that special relativity cannot handle accelerating objects or accelerating reference frames.

Inertial frames only

Inertial Frames

An inertial frame is a system of spatial coordinates in which the Law of Inertia holds. In other words, an inertial frame of reference is one that is not accelerating. Any reference frame that moves with constant velocity relative to an inertial frame Is an inertial frame.

Acceleration

Is Allowed

Needs General Relativity

In these situations, the theory of General Relativity is called for. Velocities are relative but acceleration is treated as absolute. To accommodate this change, general relativity has to be used in curved space-time.

But we are not here to follow these cumbersome arguments. There are plenty of them published in the internet. We will only present the final outacome in an easy format.

Not transformable

In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is regarded as an absolute entity. So acceleration is not qualified for this inertial club, and is supposedly out of reach by the special theory, or, not until the theory of General Relativity is employed.

Acceleration Frame Transformation

Acceleration Transformable

Many relativists believe that this is a common misconception that special relativity cannot handle accelerating objects or accelerating reference frames is not true. Although the theory of Special Relativity treats accelerating frames differently from inertial frames, it can still deal with accelerating frames.

Lorentz Transformation - Again

Again, the relativist has to resort to the same old trick of employing the Lorentz transformation regardless of the initial inspiration of Special Relativity.

Transformation Equations for Acceleration

𝑎1𝑧 =𝑑𝑣1𝑥

𝑑𝑡1

=𝑑𝑣1𝑥

𝑑𝑡2

𝑑𝑡1

𝑑𝑡2

𝑣2𝑥 + 𝓋

1 + 𝑣2𝑥𝓋/𝑐2𝛾 1 − 𝓋/𝑐2 𝑣2𝑥

=𝑎2𝑥

𝛾3 1 + 𝑣2𝑥𝓋/𝑐2 3

Paul Lorrain & Dale Corson: Electromagnetic Fields & Waves. W. H. Freeman & Company.1970. 2nd Ed. P. 217.

The Six Transformation Equations

𝑎1𝑥 =𝑎2𝑥

𝛾3 1 + 𝑣2𝑥𝓋/𝑐2 3

𝑎1𝑦 =1

𝛾3 1 + 𝑣2𝑥𝓋/𝑐2 3×

𝑎2𝑥 −𝑣2𝑦𝓋

𝑐2 + 𝑣2𝑥𝓋𝑎2𝑥

𝑎1𝑧 =1

𝛾3 1 + 𝑣2𝑥𝓋/𝑐2 3×

𝑎2𝑧 −𝑣2𝑥𝓋

𝑎2𝑥 =𝑎1𝑥

𝛾3 1 + 𝑣1𝑥𝓋/𝑐2 3

𝑎2𝑦 =1

𝛾3 1 + 𝑣1𝑥𝓋/𝑐2 3×

𝑎1𝑦 +𝑣1𝑦𝓋

𝑎2𝑧 =1

𝛾3 1 + 𝑣1𝑥𝓋/𝑐2 3×

𝑎1𝑧 −𝑣2𝑧𝓋

Paul Lorrain & Dale Corson: Electromagnetic Fields & Waves. W. H. Freeman & Company.1970. 2nd Ed. P. 217.

Out of the Pandora’s Box

So instead of the initially defined precinct of inertial frame, the tentacles of Relativity is now stretching out of the inertial box. The transformation of acceleration may be regarded as one of the major attempts of Relativity encroaching into the domain of classical physics. We can see how it is leading classical physics further and further astray.

ACCELERATION IN VISONICS

Cosmic Adventure 5.12c

Apparent Velocity

= Actual Velocity

In our investigation into the observed speed of an object, we found that the

apparent velocity and the actual velocity are the same. So it is easy to

work out with acceleration.

Apparent = Actual

By definition, acceleration is the change in velocity. So by differentiating the velocity against time we have the apparent acceleration as:

𝑎2 =𝑑𝑢2

𝑑𝑡=

𝑑𝑢1

𝑑𝑡

= 𝑎1

Much simpler than those equations in Relativity.

Apparent Acceleration =

Actual Acceleration

Visual Observation cannot tell

The result is: The observed acceleration is the same as the actual acceleration.

This is not surprising as the change in space nullifies the effect of change in time, same way it does with velocity.

Another reason is: acceleration is an internal change which cannot be visually detected, and we shall see.

The treatment of visual effect in visonics is nothing new. It is only an extension of classical physics into optics involving the speed of light. Classical optics treats the speed of light as infinite. Einstein treated light as a bird caged in his magic of frame transformation. Modern optics is even worse, it incorporated the concept of waves with Einstein’s relativity theory.

The Pivotal Role of Acceleration

But acceleration is a very special motion in physics because the cause of acceleration is apparently the key to the nature of motion.

Acceleration plays a pivotal role in all the dynamical quantities such as time, velocity, force, and mass in classical and modern physics:

1. Acceleration in time gives rise to a velocity:

𝑎𝑡 = 𝑣

2. Change in velocity defines acceleration:

𝑎 =𝑑𝑣

𝑑𝑡

3. Force is the product of mass and acceleration:

𝐹 = 𝑚𝑎 =𝑚𝑑𝑣

𝑑𝑡

Acceleration in Various Situations

Nature of Acceleration

However, the nature of acceleration is not well understood. What the best Isaac Newton could do is to define force as the change of momentum in time and acceleration as the change of velocity in time. As to what the nature of force, momentum, and acceleration is, he could not have a clue. So the classical mechanical system he set up stops short at only the definitions of the famous three laws.

Modern Ideas

Not even great geniuses like Einstein could explain what acceleration is. That was why he was led to the notion of space-time distortion to account for gravitational force and acceleration, thus leading physics astray for over a hundred years

Modern physicists offers a plausible answer by the concept of particle exchange. This is a partial solution, not the complete and true one.

Comparisons

But before we venture into the wonderful domain of acceleration, let us sum up what we have been discussing so far. Let us made a clear picture of our present situation by way of comparison of the concepts of classical physics, theory of Special Relativity and visonics.

It is necessary because of way relativity encroaching into the realm of science is like a religion spreading throughout the cultured society. It will need endless persuasions and arguments, or even wars, to put it in its right perspective.

However, once we establish this easily recognizable comparisons, we shall create a clear way ahead for all coming discussions.

COMPARISONS

To be continued on Cosmic Adventure 5.13

CA 5.12 Acceleration Transformation

Science