Wikipedia:Reference desk/Archives/Mathematics/2006 November 8

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November 8

Numerology

I am working on a school project, which counts as half my grade. The project is on numerology. We were each assigned a number and we have to research at least 30 facts about our number. My number is 3 and i need more facts!

Try our article on the number three. Philbert^2.71828 04:28, 8 November 2006 (UTC)

One property of 3 is that you can't get a better grade than that if you don't do this project and it counts as half your grade. – b_jonas 09:51, 14 November 2006 (UTC)

what leval are you working at? No point in answeres that are for Kindergarden kids if you at Grad School and vice versa (extreem example, but you get what I mean)

What logical fallacy is this?

a --> b
c --> b
ergo a --> c

Who would ever think that was true ? That's like saying:

• If it rains the street will get wet.

• And, if I hose down the street it will get wet.

• Therefore, if it rains I have hosed down the street.

StuRat 05:31, 8 November 2006 (UTC)

I'm not sure that it has a name per se, it seems to be an example of Affirming the consequent (sometimes called "assuming the converse") together with one correct step.

(1) a --> b
(2) c --> b
(3) from (2) (fallacious step) b -> c
from 2 and 3 a --> c (valid given lines 1 and 3). JoshuaZ 05:43, 8 November 2006 (UTC)

just some fyi for myself

if i have an object that weights 36 pounds moving at 5 ft per second. what would be the weight at sudden impact. so, say it was a 36 pound ball dropped from 5 ft and say it took a second to hit the ground. what would be the impact weight?please help if possible.

Rosovyn 04:27, 8 November 2006 (UTC)

Do you mean impact force ? StuRat 04:45, 8 November 2006 (UTC)

Or it could be an impulse I suppose if the contact time was very short. Impulse is the same as momentum. Q doesnt say whether ball bounces. Assume it doesnt. Force = rate of change of momentum. Hey presto!--Light current 04:54, 8 November 2006 (UTC)

To clarify, Light Current is saying that the force ("weight" at impact) can be arbitrarily large, depending on how quickly the object comes to rest (or bounces, or whatever). Dropping it through Jell-O would stop it with very gradual force, whereas dropping it onto steel would create a very sudden violent force. What you do know is that the total work and impulse is the same up to the extra falling distance that the object gets if it sinks significantly into the surface (as for Jell-O and not steel). --Tardis 18:37, 8 November 2006 (UTC)

An object dropped from a 5 ft height will hit the ground in slightly more than half a second with a speed of approximately 18 ft per second. Unless you fiddle with the acceleration due to gravity, you cannot drop an object from 5 ft and have it take a full second to hit the ground. That corresponds to an acceleration of 0.31 g, a bit less than the gravity on Mars. With that acceleration, the speed on hitting the ground is 10 ft per second.  --Lambiam^Talk 06:14, 8 November 2006 (UTC)

This question is more about physics than mathematics; so even though we're used to that, our answers may be inferior to those available at the Wikipedia:Reference desk/Science.

What we can say is that an object with a given mass (preferred to weight) falling through a constant gravitational field (assumed that on Earth's surface) for a given distance will have a definite momentum. Momentum would be calculated as mass times velocity. Since gravitation constantly accelerates the object, the farther it falls the greater will be its velocity, and thus the greater will be its momentum on impact.

However, although momentum is one way to say "how hard it hits", it's not clear how that relates to your "impact weight".

For example, a somewhat unlikely interpretation invokes special relativity. Acceleration increases the mass of an object; but this effect is almost too small to measure in your experiment.

Part of learning physics is learning to use its vocabularly correctly. We give familar words precise technical meanings. Thus we distinguish between mass and weight, between velocity and speed, and so on. Words like force and work have common English meanings, which now only confuse us.

What others are suggesting is an idea other than momentum (or relativistic mass!). It takes a certain constant acceleration to achieve a given speed over a given distance. The greater the distance, the less acceleration is required to achieve a fixed velocity. The same reasoning applies to stopping the object. If it comes to a halt over a given distance, it is "decelerating", though physics just calls this acceleration in the opposite direction. If the distance is small, the (negative) acceleration must be large; if the distance is larger, the stopping acceleration is less.

Larger acceleration implies greater force. This is Newton's famous law, F = ma; force is directly proportional to acceleration (and to mass). Thus when an automobile stops by hitting the side of a building, the forces on the driver can be fatal; but when it stops by using its brakes, all is well. Either way we have the same momentum (mass times velocity) to dissipate, but the crash involves much greater force than braking. --KSmrq^T 22:36, 8 November 2006 (UTC)

admissions problem

ok, I'm applying to n number of schools. I'm an average candidate and the admissions rate at each school is 10%. What is the probability of getting into exactly one school ?

C(n,1)*(9/10)^(n-1)*(1/10) right!? ?

so why does is above equation for n = 10 greater than n = 15?

if I changed the problem to "What is the probability of getting into AT LEAST one school ?"

and then added up C(n,1)*(9/10)^(n-1)*(1/10)^1 + C(n,2)*(9/10)^(n-2)*(1/10)^2 + ... + C(n,n-1)*(9/10)^1*(1/10)^(n-1) + C(n,n)*(9/10)^0*(1/10)^n

(is that last term correct?!?)

then would n = 15 be greater than n = 10 ? (just did it in Excel, .794 > .651, still not sure if this is right though)

For the first question: Imagine you applied to 78329432327983193 schools. What are the chances of being admitted to exactly one school, and being rejected by all 78329432327983192 other schools. Pretty low, no? Most likely, you will be accepted by 7832943232798319 schools in total. Now if you apply to (only) 15 schools, the expected number of admissions is 1.5, which means there is a fair chance two or more schools will accept you. So what you see is the chance of more than one accepting school kicking in, which diminishes the probability for exactly one.

For the second question: Yes, that's correct, but there is a much simpler way of getting there. What is the probability of being rejected by all n schools? It is (9/10)^n. The complement 1–(9/10)^n is the probability of not being rejected by all n schools; in other words: of getting into at least one. To see that this formula is equal to your formula, check out the binomial theorem.  --Lambiam^Talk 10:28, 8 November 2006 (UTC)

Lambian, why do don't I need to use any C(x,y) stuff for your method??

Why should you if it works without? If you have a one in a million chance of winning the lottery, you have a one in a trillion (million times million) chance of winning the lottery twice out of two tries. See, no C! But if you insist, (9/10)^n = C(n,0)*(9/10)^n*(1/10)^0.  --Lambiam^Talk 15:10, 8 November 2006 (UTC)

I don't see why your chances to be admitted to a specific school should be 1/10. Schools typically don't pick students at random.--gwaihir 12:22, 8 November 2006 (UTC)

Good point... the questioner has clearly looked at the number of applicants, and the number of acceptances, and seen that typically one tenth of the applicants get accepted. So, they choosen to model their application process by assuming that it is random. Yes, it isn't the same as the real world (as you point out, universities don't pick students randomly as far as we know), but it forms a good approximation. That's the whole point of a matematical model - it simplifies things, accepts that the answer won't be exact, but tries to ensure that the answer is close to the truth. Tompw 12:09, 14 November 2006 (UTC)

What sequence satisfies this constraint?

Hi. For a computer science project, I've come across a sequence of variables, which I have proved satisfies the following constraint:

x1+x2 > x1+x3 > x2+x3 > x1+x4 > x2+x4 > x3+x4 > x1+x5....

(All variables are reals)

I've tried to set xi=n-i for some fixed n, which fails even if we allow \leq instead of < (which is acceptable to me).

n=11:

19>18>17>17>16>15>16... as x1+x5=(10+6)=16, but x3+x4=8+7=15.

I am guessing perhaps xi=1/k * x{i+1} would satisfy this, or maybe something similar would. I've tried hard, but not been able to prove anything.

Does such a sequence even exist? My project would be A LOT easier if someone could prove that it does not.

Thanks in advance

Søren 130.225.96.2 13:38, 8 November 2006 (UTC)

$x_{n}=-2^{n}$ --gwaihir 13:54, 8 November 2006 (UTC)

Ahh, sorry, I forgot to specify that $x_{i}>0$. Otherwise that would have been a good solution. I know my example $x_{i}=n-i$ was not very helpful in this regard. Søren 130.225.96.2 14:09, 8 November 2006 (UTC)

I gather that in general $x_{n}+x_{n+1}>x_{1}+x_{n+2}$. You can show then by natural induction that $x_{n+1}<x_{1}-F_{n}(x_{1}-x_{2})$, where $(F_{n})_{n}$ stands for the Fibonacci sequence, with $F_{0}=0$, $F_{1}=1$. Since $F_{n}$ grows unboundedly, for some index it will exceed $x_{1}/(x_{1}-x_{2})$, forcing $x_{n+1}$ to be negative.  --Lambiam^Talk 15:02, 8 November 2006 (UTC)

Slight simplification: $x_{n-1}+x_{n}>x_{1}+x_{n+1}$ implies $x_{n}-x_{n+1}>x_{1}-x_{n-1}\geq x_{1}-x_{2}$ for $n\geq 3$, but this contradicts $x_{n}>0$ for large $n$.--gwaihir 15:05, 8 November 2006 (UTC)

Thanks, that was a very nice solution - Søren 130.225.96.2 07:29, 9 November 2006 (UTC)