Physics, despite it's reputation is an amazing science and with not as much effort as you would expect you can make calculations on some very interesting/practical phenomena. In this instructable I intend to show you how to make and understand some of those calculations, a minimum of algebra is necessary but the rest i will explain here. Please note this is my first instructable, more will follow on topics such as basic electronics ( a simple electric shock device) and animation by computer ( using blender).
Step 1: Algebra
Maths is crucial for solving the problems that will follow, you will need to understand standard index form numbers where you have Zx10^y this will give you a long number starting with z followed by y zeroes( 2.99x10^8=299000000), also you will need to be able to rearrange equations such as E=mc^2 into m=E/c^2 and more complex examples such as newton's law of universal gravitation. To do that for example if we wanted to find the mass of an object from already knowing what it's force upon us was, what our mass is and how far away it is we would do something like this. M is the objects mass, m is our own mass, G is a constant 6.67x10^-11 and r is how far we are from the object
F r^2 = GMm
F r^2 / Gm = M
this new equation lets us put in the numbers and get the answer out. Do not fear letters in maths!
Step 2: First Look at the Problem, Work Out What It Is Asking For.
Start by taking a close look at what quantity you want to find and which ones you already know, some data such as constants might not be immediately obvious however by their very nature constant remain the same whatever happens ( or so we currently think). Newton's constant, the gravitational constant, is always 6.67 x10^-11 N m^2 kg^-2( it has this confusing unit because it must multiply by 2 masses in kg and divide by a distance in metres squared as given by the universal law of gravitation, you can ignore this for now more on units later) and therefore to use the universal law of gravitation we only need to know the masses of objects and the distance between them. If you find that the numbers you have do not match any of the quantities in the formula you have you will need to either use a different formula if you can find one or to use a formula to convert the quantities you have into those you want. for example if i knew an object's ( lets make it a tank) weight on earth and wanted to know how much force it would exert on a 1 kg mass if they were alone together far away from any other massive objects I would start by dividing the weight of the tank by the force per mass on an object at earth's surface to get the mass and then use newtons gravitational law to find the force using the mass I have found. Or if i knew what acceleration a known force produced on this tank i could use f=ma to divide the force by the acceleration and get the mass.
Step 3: Rearrange the Formula
When you know what formula to use you need to rearrange it so that the quantity you want is alone on one side of the equals sign and the rest of the stuff is on the other. This is very easy to do I will show you how starting with V=s/t
To get v we divide s by t. We can swap numbers across the equals but must change their sign when we do so, this means where we have + it becomes - on the other side and vice versa. Where we have multiply it becomes divide, where we have square ( ^2) it becomes square root.
so v=s/t becomes vt=s .
Then i'll show you a harder one, if you can do this easily i doubt you will have any problems rearranging most of the equations you use in physics.
v^2=u^2+2as so that a is on it's own( the subject). Answer on next page.
Step 4: Put in the Numbers
Now we have the correct formula for the job we put in the quantities here a a few common letters and what they correspond to
c=speed of light
g=gravitational force per kg at earth's surface/ acceleration due to gravity at earth's surface
Now i know some of these numbers are the same but it will be quite clear which ones an equation needs, such as v is voltage when using electric currents but velocity when talking about motion
P can be momentum or P can be the power of an electric current, when momentum and power are in the same equation power will often be represented by W( for watts the unit of power). Just drop the numbers into the formula, do the calculation and out comes the answer.
The answer to the question at the end of the previous page is a=(v^2-u^2) / (2 s).
Step 5: Units
So far i have explained about equations and formulas but i have not mentioned how we measure them, yes it's alright that E=mc^2 but that means that if we do not use standard units for the energy, mass and speed of light two people can get completely different answers. Alice measures a mass to be 45000kg and c to be 3x10^8 ms^-1, she gets a mass energy of 4.05 x10^21. Bob also performs the calculation he uses a mass of 45 metric tonnes and a speed of 671 million miles per hour, he gets an energy of 2.02x10^19. they can't both be correct. To avoid this we use particular units for certain quantities like kilograms for mass, ms^-1for speed and velocity, Newtons for force, kgms^-1 for momentum, volts for voltage, amperes for current, ohms for resistance. If a quantity you are using is given in kilo or mega units multiply by 1000 or 10^6 respectively. For milli units divide by 1000, when using kilograms this does not apply as the kilo unit is standard not the gram however most masses are given in kg, grams or metric tonnes. This way Alice and Bob must use the same numbers for the same object and get the same answer which is given in joules.
Step 6: More on Units
Quite often you can have quantities,especially constants given in very strange units, this is because the units must cancel out to make the units on the other side of the equation. Also we can sometimes almost find the equation just from knowing what the units are for example knowing that newtons can be written as kg m s^-2 we that momentum which is in kg m s^-1can be divided by seconds to get force. This turns out to be true as F=(change in momentum)/ (time for change to happen). Also amps(C s^-1) multiplied by volts(J C^-1) cancels to remove the C from the units and give joules per second ( power). We can use this to deduce that planck's constant found in the equation E=hf must be in J s because for joules to be the units on the left hand side we must find something that will cancel with hertz ( this is the unit for freqeucy f and can simplify to s^-1), J s will do this even if it seems a strange idea.
Step 7: Deriving Equations
Now we reach the hardest stuff i will cover in this instructable , how to fit equations together to make new ones. This is done by either substituting in a second equation to make one of the quantities in the original or by finding equations that both produce the same quantity, such as energy. I will show you what i mean here.
If we want to know how fast an object of any mass will be going when it hits the ground after being dropped we can say that by the time it has fallen distance h it's kinetic energy will be equal to the gravitational energy it has lost because energy is always conserved.
so E=0.5mv^2 and E=mgh
cancel the m to give 0.5v^2=gh
are you starting to recognise this?
It turns out to be V^2=U^2 +2gh if we include an original velocity up or down at the time it was dropped. Remember g is acceleration and h is displacement, we've produced V^2=u^2 +2as from using energy and gravity!!!