## Background

### Velocity

A definition of velocity is *the speed of something in a given direction (*illustrated in the picture below). Velocity is measured as a change in position over time, such as 25 miles per hour or 5 feet in a second. If velocity is constant, it’s the amount of time that it takes to travel the distance.

Isaac Newton defined an important law for velocity in the 17th century stating that an object at rest (no velocity) stays at rest unless a force is applied. Velocity and force are linked together in **Newton’s first law of motion**.

The first law also states that an object in motion will stay in motion with the same speed and in the same direction (i.e. velocity) unless a force is applied. This means that an object can change its position even if there is no force. This led to Newton’s second law, explaining a change in velocity over time, which is acceleration.

### Acceleration

Acceleration is the rate of change of velocity with respect to time. For example, push on the gas pedal to accelerate a car and it changes velocity (speed in a given direction). Push on the brake, and it decelerates and will decrease velocity. If the car hits another object and decelerates, the force of the impact is based on the acceleration or deceleration, not the velocity of the car.

**Newton’s second law of motion** states that the sum of all forces (F) is equal to mass (m) multiplied by acceleration (a), or F=ma. Thus, acceleration and force are intimately linked with this law.

Velocity is a change in position over time. Acceleration is a change in velocity over time.

## Explanation

### Velocity and Newton’s First Law of Motion

The first law of motion states that a body will remain at rest, or continue at a constant velocity, unless a force is applied. Essentially, velocity is always constant in this law. At rest, velocity remains zero. While in motion, velocity remains the same until a force is applied. At rest, a particle or body’s acceleration (a) is zero and velocity (v) is zero. The figure below describes a particle that has a particle core of one or more wave centers, a standing wave structure that extends to the particle’s radius, and spherical, longitudinal traveling waves beyond this radius. Standing waves are generated by in-waves that are reflected to become out-waves. At rest, the wavelength of the standing waves (λ_{lead}) matches the wavelength of the in-waves (λ_{l}). There is no wavelength/frequency difference between the particle and its surrounding environment. The particle is at rest and will remain at rest.

**Note: **The traveling wave is a spherical, longitudinal wave. The figure above illustrates a simple sine wave as this wave due to the difficulties describing a three-dimensional wave in a two-dimensional image.

A particle in motion has a velocity greater than zero. With no acceleration (a=0), velocity remains constant according to Newton’s first law of motion. To an observer, the wavelength on the leading edge (λ_{lead}) of the particle is compressed, in the direction of motion, relative to the longitudinal, traveling in-waves (λ_{l}). There is a wavelength or frequency difference with the external environment, which is shown to follow Doppler equations. This frequency difference is the basis of the calculation for velocity, as it will be shown mathematically. It is also the reason particles (and thus objects that are built upon particles) experience time dilation and length contraction in the direction of motion.

The figure below illustrates a particle in motion with no acceleration. Wave amplitude is constant and equal on all sides of the particle. The particle will maintain its standing wave frequency on both the leading and lagging edges of the standing wave structure, although it is different than its external environment. Wavelength on the leading edge is less than the wavelength of the in-waves (λ_{lead} < λ_{l}) and will remain constant. The smaller the leading edge wavelength, the greater the velocity. If the particle core reaches the edge of its standing wave radius, the leading edge wavelength is near zero, and its velocity is nearly the speed of light.

Velocity results in a difference in wavelength. The Doppler effect is apparent in the image above of a moving particle and is the basis of how the Velocity Equation is derived. The ratio of the fundamental wavelength and the wavelength of the leading edge of the particle is defined as:

Since the speed of light (c) and the fundamental wavelength (λ_{l}) are both constants, velocity (v) is proportional to the leading edge wavelength (λ_{lead}). It will be described later below on this page as an equation for velocity.

### Acceleration and Newton’s Second Law of Motion

In the first law of motion, when described in wave terms, wavelength may differ but wave amplitude is constant. In the second law, wave amplitude is not constant and is the reason for force and acceleration. Particles move to minimize amplitude, one of the laws of energy wave theory. The figure below describes a particle, initially at rest (v_{i}=0), now accelerated (a>0). A particle will move in the direction of minimal amplitude, thus the leading edge in the acceleration and velocity equations always refer to the direction in motion where wave amplitude is less than the amplitude in any other direction surrounding the particle. Similarly, the amplitude in the direction of motion is described using the same notation (A_{lead}). This amplitude difference leads to a force on the object and it is accelerated. The particle core will move from its resting position to a new frequency.

The next figure describes the change in position of the particle core due to acceleration, which is caused by amplitude difference. Acceleration is the change in the position of the particle core, affecting wavelength and frequency. A smaller acceleration value moves the particle core slower towards its edge, and a larger acceleration value moves the particle core faster towards its edge. This requires a measurement of time to define *slower* or *faster*. Time is based on wavelength cycles (which is frequency), thus acceleration is the movement of the particle core towards the edge of the radius based on the number of wavelength cycles. This is seen in the Acceleration Equation.

### Newton’s first law of motion is based upon wavelength. The second law is based upon wave amplitude.

## Equation

### Velocity

The velocity equation is found below. The detailed derivation is found in Section 6 of the *Forces* paper. The equation can be used to model any group of particles (Q) at distance (r) at time (t), assuming constant acceleration. Note that this simplified version assumes a particle begins at rest.

**Velocity Equation**

In the calculations, velocity calculations were compared to known results of the velocities of falling objects on Earth. Gravity is a form of constant acceleration, at least when measured close to the surface of a planet. For example, calculations for gravity on Earth are normally calculated at 9.81 m/s^{2} – constant acceleration. Thus, velocity was cross-checked with velocities of falling bodies due to gravity to validate the equation. However, the equation requires a modification for the amplitude factor for gravity, as gravity is a slight loss of wave amplitude. The equation is modified to have this factor. Q is the particle count for large bodies and α_{Gp} is the amplitude difference for nucleons; both are described in the section on gravity. μ is the proton-to-electron mass ratio.

**Velocity Equation for Gravity**

### Acceleration

The acceleration equation found as a component of the Force Equation. The other component is mass. The equation is:

**Acceleration Equation**

Acceleration calculations were compared to known results of the surface gravities (accelerations) for planets. The equation is modified similar the velocity equation for gravity.

**Acceleration Equation for Gravity**

Detailed steps to derive the equations are found in the *Forces* paper.

## Proof

Proof of the energy wave explanation for particle motion (velocity and acceleration) is the derivations and calculations of:

- Newton’s Second Law
- Calculated the velocities of falling bodies on various planets (at varying times) –
*see example calculation below* - Calculated 11 planet surface gravities (acceleration) –
*see example calculation below*

### Example Calculation – Velocity

Earth – Velocity of Falling Body after 50 seconds from rest: Velocity is based on time, so the first calculation is based on a falling object on Earth, which starts at rest, and has a calculated velocity after 50 seconds (t). Using the Velocity Equation for Gravity above, the number of nucleon particles for Earth was calculated is used as Q_{group} (refer to the Gravity section for calculation of nucleons). The radius of the Earth, r_{group}, is 6,375,223 meters. The proton to electron mass ratio is μ = 1836.152676.

t = 50 s

r_{ earth} = 6,375,223 m

Q_{earth} = 3.570 x 10^{51}

**Calculated Value: **490.33 m/s

**Difference:** 0.000%

### Example Calculation – Acceleration

Earth Surface Gravity: The Acceleration Equation for Gravity is used to calculate the surface gravity for Earth. The number of nucleon particles for Earth was calculated and is used as Q_{group }(refer to the Gravity section for calculation of nucleons). The radius of the Earth, r_{group}, is 6,375,223 meters. The proton to electron mass ratio is μ = 1836.152676.

r_{ earth} = 6,375,223 m

Q_{earth} = 3.570 x 10^{51}

**Calculated Value: **9.807 m/s^{2}

**Difference:** 0.000%

* Note: *A summary of various velocity calculations and acceleration calculations are found on this site; more detailed calculations with instructions to reproduce these calculations is found in the

*Forces*paper.