Ah, fractions. The cornerstone of 4th grade math. Why is it so hard to get our little heads around fractions? In math a fraction is simply a way of describing a part of a whole. The Latin origin is helpful: the word fraction entered English in the 14th century from the Anglo-French word fraccioun, which came from the Old French fraccion. The Late Latin word fractionem meant a breaking and came from fractus, the past participle of frangere meaning to break something into parts. Here is where it gets interesting. Yes, the English words fragment and fracture share the same roots, entering English around the same time. The Proto-Italic root *frang- came from the Proto-Indo-European *bhreg- meaning to break (and root for the word break) but also Old English brecan meaning to break, Lithuanian brasketi meaning to crash or crack and Old Irish braigim meaning to break wind. Yes, all things 4th grade eventually make their way back to breaking wind. A fraction, then is just a part broken off from a whole. That wasn’t so hard, was it? Now, who farted?
This post is best viewed on my blog, click here to be taken to its permalink.
Different problems involving kinematics often share common properites which can be used (or not used) depending on the scenario. Once you become more adept at solving these problems individually, you’ll begin to easily recognise situations where some conditions apply and others may not and start to adapt the conditions to fit the scenario. A bit of thought is required to decide logically whether one condition may be physicaly viable in the situation or not.
In this post, I will look at topics that are not strictly kinematics but have applications within the field and in wider mechanics.
There are some common boundary conditions (i.e. what happens at the extreme-most points of the motion) which come from logical reasoning.
For motion of an object with its weight perpendicular to flight motion it can be assumed that the x-component of velocity stays constant throughout the flight. This means that the final xvelocity, vx, is equal to the initial x velocity, v0, x.
Themaximum height of flight, ymax,is always the point where the y
velocity, vy = 0. This is
a direct result of calculus, wherein dy/dt = 0 at a turning point (a
maximum or minimum of a function). Now, if the time to reach the maximum height
is equal to the time to reach the ground again (i.e. for perfect parabolic
flight where the launch height = landing height) then we can assume that the time
to reach ymax is equal to
half of the full flight time.
if we set the original position of the object at the origin of our system of
co-ordinates, the point at which we find it at the same height again (the point
where it lands, assuming it lands at the same height) by setting y = 0.
something is accelerating then its vectorial sum of all forces is equal to ma,
where a is its resultant acceleration. If it is in equilibrium (“stationary” or “moving at constant velocity”) then the
vector sum of its forces is equal to zero. This is a very important consequence and can be used extensively to solve problems.
should always remember to resolve all forces/vectors in a free body
diagram when working on the x and y components separately. Unless
a vector is at 90° from the direction you’re
resolving it to it will contribute that component. For example, the vector W acts only in the y-direction, meaning
we can leave it out of our x-component
if a vector points in the opposite direction to the direction you’re working
in, give its magnitude a negative sign. This is what’s intended by the vectorial (or algebraic) sum as opposed
to a simply scalar summation.
in kinematics we ignore the effects of resistive forces – especially air
resistance – since they can make problems more complicated to solve. However,
resistive forces such as the friction force Ff or the reaction force R can be used to our
The magnitude of the frictional is given by
Ff = μR,
is the friction coefficient. Vectorially, we define Ff to be perpendicular to the reaction force R (along the surface of the material) and
opposite to the direction of motion with unitary vector v. Hence we can express
Ff = − μR v,
the magnitude will usually suffice.
shall further examine what is meant by the reaction force. The reaction force
is defined in a direction perpendicular to the surface the object rests upon
and will often have its y-component Rx equal in magnitude and
opposite in direction to the weight W
of the object, assuming the weight acts purely in the y-direction. See Newton’s third law of motion for more information.
Thus we have,
Rx = W,
the object is fully supported by the surface it sits upon.
results can be useful when applied simultaneously. If we know the frictional force
and friction coefficient, we can find the reaction force and, in turn, the
object’s weight. Since we know that
I should probably stop now. It’s 11:20 pm and I’ve checked 10 out of 18 of the things I had to do this weekend. Today I studied:
• History (Middle Ages)
• Chemistry (Periodic Table properties)
• a bit of the SAT’s writing section
I guess it was a pretty productive day!
oh and I downloaded that tree app and it’s so good that I actually felt bad for killing the tree.
/February 28th 2015/
I think that mathematically interested sixth formers, wanting to study the subject at University, should have some background knowledge. Many would have heard about the Riemann Hypothesis and a link to prime numbers; this subject is beyond the scope of this blog but school pupils, with knowledge of prime numbers and geometric progressions, can see how the Zeta Function relates to prime numbers in the above bit of mathematics. I like it anyway.