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The '''Bully Mnemonic Extension''' is a technique for remembering the exact number of meters that light travels in one second, and the approximate range of gravitational accelerations that occur on the surface of the Earth due to Newtonian gravity. The ''Bully Mnemonic Extension'', when used in conjunction with the ''Bully Mnemonic'', allows one to calculate a number of physical quantities, including the exact number of meters in a light year.


The following relationships are encoded in the Bully Mnemonic Extension:

<math display="block"> Speed \, of \, light \, in \, vacuum = {299,792,458 \frac{m}{s}} </math>

<math display="block"> High \, end \, of \, gravity \, on \, Earth's \, surface \approx {9.86 \frac{m}{s^{2}}} </math>

<math display="block"> Low \, end \, of \, gravity \, on \, Earth's \, surface \approx {9.76 \frac{m}{s^{2}}} </math>

The following relationship can be derived using the ''Bully Mnemonic Extension'' in conjunction with the ''Bully Mnemonic'':

<math display="block"> 1 \, light \, year \approx {9,460,528,400,000,000 \, meters} </math>


= Bully Mnemonic Extension Steps =

== Initial Definitions ==

=== Step 1 ===

Complete steps 1 and 2 of the [[The Bully Mnemonic]] to form integers a) and b) as shown below:

<math display="block"> \begin{matrix} {\color{Red} 1} & \scriptstyle\text{2} & {\color{Red} 3} & \scriptstyle\text{4} & {\color{Red} 5} \end{matrix} </math>

<math display="block"> a) \, {\color{Red} 1} 0  {\color{Red} 33}  0 </math>
<math display="block"> b) \, {\color{Red} 3} 0  {\color{Red} 55}</math>

=== Step 2 ===

The Bully Mnemonic Extension will use two variants of integer a). The first variant will have 33 removed and replaced with 00.  The second variant will have 330 removed and replaced with 22:

<math display="block"> av1) \, {\color{Red} 1} 0000 </math>
<math display="block"> av2) \, {\color{Red} 1} 0  {\color{Red} 22}</math>
<math display="block"> b) \, {\color{Red} 3} 0  {\color{Red} 55}</math>

== Speed of Light ==

=== Step 3 ===
Multiply integers av2) and b) from Step 2.

<math display="block"> {\color{Red} 1} 0 {\color{Red} 22} \times {\color{Red} 3} 0 {\color{Red} 55}  = 3122210 </math>

Using Long Multiplication: 
      3055
 ×    1022
 ————————————
      6110
     6110
    0000
   3055
 ————————————
   3122210

=== Step 4 ===

Drop the zero from the integer obtained in step 3, swap each 2 with 3, and swap each 1 with 9, to obtain integer f) shown below:

    312221
 f) 293339

=== Step 5 ===

Multiply integer av2) from Step 2, and integer f) from step 4, to get the total number of meters that light travels in one second.

<math display="block"> {\color{Red} 1} 0 {\color{Red} 22} \times 293339  = 299792458 </math>

Using Long Multiplication: 
          1022
 ×      293339
 ——————————————
         9198
        3066
       3066
      3066
     9198
    2044
 ——————————————
    299792458

== Gravity on Earth ==

=== Step 6 ===

Divide the speed of light obtained in step 5, by integers av1) and b) from step 2, to obtain a value for Earth's gravity:

<math display="block">{\frac{299792458 \frac{m}{s}}{{\color{Red} 1} 0000 \times {\color{Red} 3} 0 {\color{Red} 55} \, s}} = {\frac{299792458 \frac{m}{s}}{30550000 \, s}} \approx {9.81 \frac{m}{s^{2}}} </math>

In terms of Long Multiplication, 30550000 and 9.81 are approximately related to 299792458 as follows: 
    30550000
 ×         9.81
 ————————————
      305500.00
    24440000.0
   274950000
 ————————————
   2997.....

=== Step 7 ===

The range of gravitational accelerations that occur on the surface of the Earth, due to Newtonian gravity, can be approximated by repeating step 6 with integer av1) increased or decreased by half a percent:

<math display="block"> {\frac{299792458 \frac{m}{s}}{({\color{Red} 1} 0000 + 50) \times {\color{Red} 3} 0 {\color{Red} 55} \, s}} \approx {9.76 \frac{m}{s^{2}}} </math>

<math display="block"> {\frac{299792458 \frac{m}{s}}{({\color{Red} 1} 0000 - 50) \times {\color{Red} 3} 0 {\color{Red} 55} \, s}} \approx {9.86 \frac{m}{s^{2}}} </math>

=== Step 8 ===

The total number of seconds in a tropical year, as calculated in step 5 of the ''Bully Mnemonic'', is 31556926 seconds.  The speed of light in vacuum, as calculated in step 4 of the ''Bully Mnemonic Extension'', is 299792458 meters per second.  Multiple these two numbers together to get the total number of meters in a light year:

<math display="block"> 299792458 \times 31556926 \approx 9,460,528,400,000,000 </math>