math problem

[steve_m]

s

[pianistimo]

was dirt falling out of the pot at the same time? 

[steve_m]

d

[ted]

It may help you to see it in terms of a simpler example. The diagonal of a unit square is equal to the square root of two, an irrational. The circumference of a circle of unit radius is 2pi, another irrational. You probably do not have difficulty seeing that the corners of the square are exact points, yet connected by a line of irrational length, or that opposite ends of a diameter of a circle are precise points, yet connected by an arc of irrational length ?

Perhaps the issue for you is that the "line" in the problem represents time instead of distance ? For problems of this sort, and many others, time is assumed to map onto a real number line, because time and a line both have the property called continuity. A section of time contains the same infinity of "points" or "moments" if you like, as a section of line. Whether a "moment" in time happens to coincide with an rational number on the line or an irrational one does not matter.

Look at it another way. If only rational "moments" existed in a time line, time would become a collection of isolated points, admittedly as finely spaced as we liked to make them, but still an infinitely small subset of the line - not a good model of time. A continuous line is a "model" of time for many physical problems.
Does this help ?

[elspeth]

Obvious question, what units did you plot your y-axis in? Because if your axis is in metres but you've plugged six feet in to work an answer out, by definition you'll get a wierd answer!

[asyncopated]

I assume that you derived the formula using the following.

Newton's law

F=m dv/dt

where F is the force, m is the mass, v the velocity and t time.  In this case, we are only worried about the downward motion of the pot, so we ignore the x and y components and concentrate on the z component of the velocity in the direction of the falling pot.

The force is that due to gravity and to all purposes on earth, we can express the gravitational force as F=mg, where g is 9.81 m/s downards

So, the equation governing the motion of the pot is

dv/dt = -g

and integrating

v = v0- gt

where v0 is the intitial velocity in parted.  We can assume the pot accidentally fell so, v0 = 0;  Also, if we want to calculate position, we need to use the definition of velocity,
dz/dt = v.

dz/dt = -g t,

and thus we have that

z = z0-1/2 gt^2

z0 can be taken as the initial height, where the post was.  The graph you see should be a parabola (y = ax^2 + b x +c). 

To answer you question, you can see that we have made a number a approximations in formulating the model.  We have assumed that the flower pot is a point like particle,  it has no air resistance, and gravity is constant at 9.81 m/s.

These are very good approximations, because even if you tried to include the effect of variation of gravity, or how the leaves of the plant moves, you still would get a similar answer to that using the simpler model stated above.  Adding detail complicates matters and doesn't add any real value in this case (often it does, but not very much here). 

As mentioned, the physical model you have used describes the world with as much detail as you want to give it.  So, really the model can't cope with the 20th decimal place in your answer.  The requirement is too accurate.  If you do need to know the 20th decimal place, use a better model.

The fact that you got an irrational number ( assuming that you have converted feet to m/s or instead used g is feet/s and measured evething in feet, but this does not matter except in getting the right answer) is really just a coincidence to the solution.  The answer is rather complicated why this is the case.  The main reason is that time was defined as a real field, and thus irrational numbers are admissible solutions.  There is nothing really strange or special about this.  It's because of the way Newton's laws where formulated. 


Having said that, irrational numbers as you pointed out are special.  One of the interesting properties is that you cannot write them out because there are in infinite number of decimal places.  Also, intoducing a real field instead of a rational field means that you have changed from a system of countably infinite number system to an uncountably infinite number system.   This is required to define the derivative d / dt. This is defined using the idea of a limit, where things get smaller and smaller so you need to do this on a real field.

Ok.  Enough of a complicated explanation for a simple thing.  Hope you find what you are looking for somewhere in this post.

[kony]

ha whats with all the caculus and Newton's f=ma stuff? useless here.

it's a fairly simple projectile motions problem (made even simpler since there is no horizontal component - i assume the pot just simply falls through the floor of the balcony.

so all you need is the famous old formula:

s = ut + (at^2)/2

where s is the vertical displacement the pot travels from the balcony to the head of the guy, u is the initial velocity (zero in this case), a the acceleration due to gravity (9.8ms^-2) and t the time.

funny, i was learning this stuff just a few weeks ago in school.

[asyncopated]

ha whats with all the caculus and Newton's f=ma stuff? useless here.

it's a fairly simple projectile motions problem (made even simpler since there is no horizontal component - i assume the pot just simply falls through the floor of the balcony.

so all you need is the famous old formula:

s = ut + (at^2)/2

where s is the vertical displacement the pot travels from the balcony to the head of the guy, u is the initial velocity (zero in this case), a the acceleration due to gravity (9.8ms^-2) and t the time.

funny, i was learning this stuff just a few weeks ago in school.

Hmm... you are right.  That's where you get the s=ut + 1/2 a t^2 formula from.  Newton's law F=ma describes how (almost) everything moves.  It tells you how to deal with other forces where you don't simply have constant acceleration (like in your case).

For example, you can't calculate the motion of the earth around the sun with your formula bacause force depends on the distance from the sun (1/r^2 force), but with newton's laws you can.   

There are two ways to learn this and most things.  Just remember the formula -- learn it the dumb way. This will get you to past most exams, but if you are really interested in the world around you and describing it, learn it the smart way, by  asking how and why?  It sticks better -- you don't forget as easily, and you take away kind of understanding which to me is an important part of living.


Anyway, I was trying to answer the initial question.  How is it possible to get an irrational number in the formulation of the model.  The reason is newton's laws, and solutions thereof are formulated such that time is a real number.  This is certainly not an obvious thing, and very far from being simple.  In fact in newton's time, when he first came up with the law, calculus and differentiation did not exist.  It's only over the 1700 and 1800s with Leibnitz, Riemann, Cantor, Lesbegue, and others that we truely understand how the machinary in the mathematics works.
 

[pianistimo]

i have nothing but awe and respect for people who simply remember the formula - let alone plugging everything in.  i just wanted to say - that if you want confirmation on answers- i found out that drexel university has a 'math hotline' or forum that you can ask questions and get answers.  that's if no one is online here at the time you need the answer.  https://mathforum.org click on the right 'join the math forum'

i found this out whilst realizing that they are part of the independent 529 plan - and if you contribute to this tax free program - you can pay for classes at the current cost as well as pick from a list of participating universities.  the only thing is that the list is kinda selective.

[kelly_kelly]

ha whats with all the caculus and Newton's f=ma stuff? useless here.

it's a fairly simple projectile motions problem (made even simpler since there is no horizontal component - i assume the pot just simply falls through the floor of the balcony.

so all you need is the famous old formula:

s = ut + (at^2)/2

where s is the vertical displacement the pot travels from the balcony to the head of the guy, u is the initial velocity (zero in this case), a the acceleration due to gravity (9.8ms^-2) and t the time.

funny, i was learning this stuff just a few weeks ago in school.

Calculus makes it easier tu understand (and, usually, to do).

[kony]

differentiating would arrive at the instantaneous velocity, acceleration, jerk etc.

whereas integrating arrive at them in the opposite order.

so.. what would you do? you are trying to find t after all.