Derivations of Formula

Below is a full list of mandatory examinable derivations as set out within the SQA AH Physics Course Specification (Version 2, 2019). In the Notes pages, all derivations (examinable or not) have been left within, to support AH Project Underlying Physics. 

Note - There is a wide range of 'Show that X is a solution of Y' questions within the course that rely only on algebraic rearrangement and substitution of  combined formula, which do not come under the heading of 'Derivations' and so have not been included here.

Unit 1 : Rotational Motion and Astrophysics 

Equation 1  -   v  =  u + at

Integration of Acceleration with respect to time :-

Equation 2  -  s  =  ut + 1/2at2

Integration of Velocity with respect to time :-

Equation 3 - Escape Velocity

The Potential Energy of an Object in a Gravitational field is given by :-

As the Gravitational Potential Energy is defined as equal to 0 J at Infinity, therefore to escape the Gravitational Potential Well, the object must be given Energy equal to :-

The lower limit to Escape completely would be to provide the object with an initial Kinetic Energy such that at infinity, all Kinetic Energy has been converted to Gravitational Potential Energy :-

Ek  +  Ep  =  0 

Equation 4 - SHM Velocity

The displacement on the vertical axis is equal to :- 

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

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

Equation 5 - SHM Kinetic Energy

The velocity of a particle undergoing SHM was found by :-

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 :-

Equation 6 - Non Reflective Coatings

When a ray of light is incident upon a non reflective coating, the ray is incident first on the magnesium fluoride coating, then the transmitted ray is incident upon the glass lens itself.

As both the surfaces have a higher refractive index, both reflected rays  have a phase change of π. 

Following the formula for destructive interference in reflected rays, it can be shown that :-

path difference = λ/2

Optical Path in Fluoride = 2 x n x d

Where n = refractive index of coating

therefore :

2nd = λ/2

If this is rearranged to solve for the thickness of the coating (d), then the following formula can be found :-

Equation 7 - Brewster's Angle

The angle of incidence  (ip) that this occurs at is known as Brewster's Angle. 

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