# Derivations of Formula

## Unit 1 : Rotational Motion and Astrophysics

### Integration of Acceleration with respect to time :-

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### Integration of Velocity with respect to time :-

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### Square both sides of Equation of Motion 1, then substitute in Equation of Motion 2 :-

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### The above formula gives the Acceleration over a large time period. If the value of θ is made to tend towards zero, the value of sinθ ≈ θ and the Instantaneous Acceleration at a point can be found :-

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### Equation 5 - Centripetal Force

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### By combining the above formulae together and by then substituting Velocity, the Period can be found as follows :-

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### However, by convention, we state that Gravitational Potential Energy is equal to 0 J at infinity, and as such , the formula is given as :-

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### The Kinetic Energy of a satellite can be found be deriving a form of Ek = 1/2mv2 , based upon a Centripetal Force calculation :-

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### The combined Energy of a satellite can be found by the addition of the Kinetic Energy and the Gravitational Potential Energy :-

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### Ek + Ep = 0

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### Equation 12 - Schwartzchild Radius (Event Horizon of a Black Hole)

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## Unit 2 : Quanta and Waves

### By differentiation, an equation of the velocity can be obtained :-

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### By differentiation of SHM Velocity, the acceleration can also be found :-

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### By combining the two equations for SHM Velocity and Acceleration and by using the trig identity below, the following can be derived :-

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### If this formula is combined with the Kinetic Energy formula, a method to find the Kinetic Energy of the particle at any given point in its motion can be derived :-

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### By substituting into the general equation above, the Potential Energy can be shown by :-

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### y = a sin 2 π ( ft - x/λ)

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### If we apply Snell's Law a formula for the Brewster's Angle can be derived :-

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## Unit 3 : Electromagnetism

### Therefore by substituting into the above Electric field strength formula :-

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### Integrating the above gives :-

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### Equation 24 - Closest Approach of Charged Particles

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### As these fields are not based upon a point source, an extended description as a base unit is required - the Force per unit length :-

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### Also, by applying calculations for Angular Velocity from the Circular Motion section of Unit 1, the Period and Frequency of the rotation can be derived :-

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