Country Boy

It sounds like you are looking for "x" such that [math]\int_a^x f(t)dt= \int_x^b f(t)dt[/math].
For example, if [math]f(x)= x^2[/math], a= 0, and b= 1, then [math]\int_0^x t^2 dt= \frac{1}{3}x^3[/math] while [math]\int_x^1 t^2 dt= \frac{1}{3}- \frac{1}{3}x^3[/math]. They will be equal when [math]\frac{1}{3}x^3= \frac{1}{3}- \frac{1}{3}x^2[/math]. Multiplying both sides by [math]\frac{1}{3}[/math] and then adding [math]x^3[/math] to both sides, [math]2x^3= 1[/math], [math]x^3= \frac{1}{2}[/math] so [math]x= \frac{1}{\sqrt[3]{x}}[/math].

Let r be the radius of the circle. Then the segment from the center of the circle to AB, with length p, the segment from the center of AB to point B, with length AB/2, and the segment from the center of the circle to B, with length r, form a right triangle with hypotenuse of length r, legs of length p and AB/2. By the Pythagorean theorem, (AB)^2/4+ p^2= r^2. Now do the same thing with chord AC getting (AC)^2/4+ q^2= r^2. Since "AB=2AC", (AB)^2/4= (AC)^2 so we have the two equations (AC)^2+ p^2= r^2 and (AC)^2/4+ q^2= r^2.

Well, I do mean offense to the OP! The original post was nothing but an attempt to get people to click on the links to advertisements. I am surpised (and a little disappointed) that this thread has gone on so long.

You often hear how Tesla is underrated and uncredited for his work.

You hear how other inventors stole all of his work and presented it as his own. You hear how he had the key to unlimited and free energy

 

It is somewhat of a mainstream opinion among non-scientists to think of Tesla as the greatest genius who ever lived.

 

I noticed that no one mentions him here and that people are more objective. It can't be a coincidence that scientists have less regard for him than regular people. It can't be because they are ''jealous of his achievements and support the system''. That makes no sense.

 

So my question is how much of this is true? Is he possibly overrated? Don't get me wrong, he was clearly a genius, but does he get more credit than he should? Is it known how many of these claims that other people stole his inventions are true? What is it that he was actually responsible for inventing but didn't get credit for it?

 

At a later portion of his life, he made claims of having discovered unlimited energy, developed lasers and VTOLs and actually claimed that he teleported an army submarine. That's an actual claim. I would guess that there is no evidence for this whatsoever. He also claimed that he can develop a weapon so powerful, that it would end all war because it would be able to destroy the whole world. Was he possibly a crackpot in his later years?

 

Anyways, there must be a reason why scientists don't talk about this other than this being a conspiracy cover-up. I would appreciate it if someone gave an objective answer.

 

I wasn't sure where to post it. It might fit in other sciences or the lounge, but seeing how we are talking about his work in physics, I thought this was the most appropriate place. If a mod disagrees, then by all means, feel free to move this.

Tesla worked for Edison with a contract that assigned all patents to Edison. I suspect whether or not one would be willing to sign such a contract depends up how much confidence one had in one's own ability to produce. It seems likely that Tesla did not have enough confidence in his own abilities to refuse to work for Edison. It is true that one of the things Edison was much better at than Tesla was "blowing his own horn"! Perhaps because of that, and the fact that he was working under Edison, Tesla does not get as much credit as he should but he certainly is well known to physicists and engineers.

As for "the key to unlimited and free energy", that's non-sense. Tesla, and Edison, were inventors and engineers, not physicist. Nothing Tesla was working on had anything to do with refuting or altering the laws of physics and "unlimited energy" would violate a basic law of physics. I hadn't read that Tesla had claimed to do all those things but they are certainly not true, especially "teleporting" anything.

Actually, an octopus does have a hard bone around its brain- but what SFN said is the best answer

Suppose that for solving for the inverse of a function that f(x) can be manipulated into the form of [math]g(f(h(x)))=x[/math]. Then, h(x) is inverted to show

[math]g(f(x))=h^{-1}(x)[/math]. Afterwards, the function is reverted to show [math]h(g(f(x)))=x[/math]. Does this correctly show that h(g(x)) is in fact the inverse of f(x)?

 

 

If not, what can be done in this situation to find the inverse of the original f(x)?

No, it doe not show that h(g) is the inverse of f. You actually have to show two​ things- that h(g(f(x))= x and​ that f(h(g(x))= x.

You can ​approximate ​an integral by a Riemann sum. Is that what you mean by "replace"?

Yes, that is true. I think that you would need to support the mirror with some backing that will not change much with temperature.

35" is pretty large for an amateur telescope. What experience do you have with making smaller telescopes?

It is also true that the cost of a telescope, mostly in the supporting structure, is roughly proportional to the cube of the diameter of the lens.

Rather than actually finding f(f(x)) you can do it this way: let y= f(x) so that f(f(x)) is f(y)= 4y- 5= 23. Solving that, 4y= 23+ 5= 28 so y= 28/4= 7. So now y= f(x) becomes 7= 4x- 5. Again, 4x= 7+ 5= 12, x= 12/4= 3. Check: f(3)= 4(3)- 5= 12- 5= 7 so that f(f(3))= 4(7)- 5= 28- 5= 23.

With samples of the man's DNA and the DNA of supposed parents, grandparents, etc. you can determine whether or not they share enough DNA to actually be "parents, grandparents, etc.". But the man's DNA alone will not "identify" his family tree. Perhaps you are referring to those "services" that use your DNA to tell you "where you are from"? Those do not determine your family tree- they simply identify parts of the world where that DNA is very common.

What do you mean by "type of media behavior"?

Assuming you are not including air resistance (which would make this problem far, far harder) the kinematic equations would be the usual [math]s= (a/2)t^2+ vt+ d[/math] where a is the acceleration vector, v is the initial velocity vector, and d is the initial position vector. Separating x (horizontal) and y (vertical) components and taking the initial speed to be "v" and the intial position to be d= (0, 0), we have [math]x= v cos(30)t=(\sqrt{3}/2)v t [/math] and [math]y= (-g/2)t^3+ v sin(30)= -4.9t^2+ (0.5)vt[/math] where v is the initial speed.

Since the ball is to end up "20 meters away, the top edge is 5 meters above the throwing point", x= 20 and y= 5.

Solve the two equations [math](\sqrt{3}/2)v t = 20[/math] and [math]-4.9t^2+ (0.5)vt= 5[/math] for t and v.

"In probability, it used to be something like: the maximum amount something could change based on a single event. However, now I can't find any evidence of that old definition."

I'm pretty darn elderly and I have never seen such a definition. The probability definition of "variance" is the sum of the square of x times the probability x occurs, [math]\sum (x- \mu)^2P(x)[/math], where x varies over all possible outcomes (and, in the case of a continuous variable, the sum becomes an integral).

Well, what are Kepler's laws?

As far back as WWII there were gliders capable of carrying small tanks. Exactly how much weight are you taliking about? And how far do you want to carry it?

Combining bleach and ​ammonia ​will release chlorine gas which is a dangerous poison. Are you sure that wasn't what you read?

I was thinking about this, and couldn't think of the answer.

 

Two objects far away from each other accelerate toward each other in space because gravity attracts them to each other. They come together to form one object. When the two objects were far apart, their combined energy was less than the newly formed object has. Where did that energy come from?

Your last statement, "When the two objects were far apart, their combined energy was less than the newly formed object has" is wrong. As Dr. P said, you need to include "potential energy".

What, exactly, do you mean by "divine"? For that matter what do you mean by "exists"? You start talking about whether the cat is "dead or alive" but then switch to "exists". I would say that once I put the cat in the box or saw someone put the cat in the box, I know it "exists" whether it is alive or dead.

Wolfram did not work because the primitive of this function is not an "elementary function" (so cannot be written in terms of trig functions, exponentals, logs, etc.).

1. Question: Rachel walks 2.80 Km to a gym at 6.00 Km/h. Upon reaching, she walks back the same distance at 7.70 Km/h as the gym was closed. Find her average velocity.

 

Attempt: I find the total time by: (2.8/6)+(2.8/7.7) ~=0.830 h. Divide 5.6 by this to get approximately 6.74 Km/h.

​

First thing I am confused by is, why is the average velocity simply not zero since her displacement over the entire time period is zero? (I actually calculated this since the back of the book gave a finite answer) Second can someone please confirm this answer?

The average velocity is 0. Your book should have said "average speed".

 

The back of the book says 3.27 Km/h but I don't know how that would be achieved

An "average" of two numbers (the "average" here is not the arithmetic average but is an average) is always between the two numbers. Since 3.27 is less than both 6 and 7.7, that is impossible.

2. Question: A car moves 0 to 900 m, starting and ending at rest. Through 225 m, it's acceleration is 2.75 m/(s^2) and through the remainder, the acceleration is -0.750 m/(s^2). What is the total travel time of the car through the 900 m?

​

 

Attempt: For the 225 m, from x-x0 = v0t + 0.5a(t^2), 225 = 0.5(2.75)(t1^2) => t1 = sqrt(1800/11) ~= 12.8 s. Find initial velocity from this for the rest of 675 m as v = v0 + at => v = 2.75sqrt(1800/11). Then from the equation for distance, 675 = 2.75sqrt(1800/11)t2 - 0.5(0.75)(t2^2), which is a quadratic and I solve it to get either approximately 66.9 s or 26.9 s neither of which help since the solution is around 55.2 (actually I had to use a solution book for this which uses slightly different values and gives 56.6 s as an answer, FYI). However, when I assume the 675 m to be covered in reverse so that the equation formulated is basically in reverse i.e. 675 = 0.5(0.75)(t2^2) this gives me t2 = sqrt(1800) ~=42.4 s, so that total time is 55.2 s.

 

So you can see why I am confused since I actually used a hunch to arrive at the answer and don't actually know what the correct method is i.e. consider the motion of the 675 m in forward with a finite initial velocity or in reverse with zero initial velocity?

Your concept is correct. With initial velocity v0 and constant acceleration a, the distance traveled in time t is indeed (1/2)at^2+ v0t. Starting from rest the car goes (2.75/2)t^2= 1.375t^2 meters in t seconds. So it does the first 225 meter is sqrt(225/1.375)= 12.8 seconds, as you say, and its speed at that point is 2.75(12.8)= 35.2 m/s. That is again what you have. The distance the rest of the way, in t seconds is given by (-0.75/2)t^2+ 35.3t meters. To complete the 900- 225= 675 m requires (-0.375)t^2+ 35.3t= 675 or (0.375)t^2- 35.3t+ 675= 0. The quadratic formula gives two positive answers, 50.6 sec or 20.0 seconds.

3. Question: A train traveling at 161 Km/h, rounds a bend and the engineer is shocked to a see a locomotive 676 m ahead traveling in the same direction, on the same track at 29.0 Km/h. The engineer applies the brakes immediately. What must the constant deceleration be if the collision is to just be avoided?

 

Attempt: 161 Km/h = 805/18 m/s and 29 Km/h = 145/18 m/s. Dl = Dt + 676, where Dl = distance covered by locomotive over time to near collision and Dt = distance covered by train over time to near collision. Replacing values according to the formula for distance covered with constant acceleration, I get the equation, (145/18)t = (805/18)t - 0.5a(t^2) + 676. Since I need to find the acceleration I assume that the train wants to reduce speed to 0 m/s and from v = v0 + at => 0 = (805/18) + at => t = -(805/18a). Then replacing this value for t in the earlier equation, I basically solve for a and get a ~= 3.91 m/(s^2).

 

Seems reasonable but it is not the answer according to the answer book and when I replace this in the first equation to get a quadratic equation to help me find time, I get t ~= 30.2 s but replacing this in the original quadratic, the sides are unequal.

 

I have tried checking where I went wrong but I cannot find the answer, not to mention that I did not have much of an idea of how to proceed with this question to being with.

Wonderful! Absolutely hilarious!

(This was a joke, right?)

Mordred referred to the more common phrasing of this so-called "paradox"- the rigid rod, rather then your string- but the explanation is the same. Because information cannot be sent faster than the speed of light, there cannot be, under the assumptions of relativity, there cannot be a "rigid rod" or a "non-extensible" string. When you pull on a string, you stretch the string and it is the stretch that is transmitted along the string. Actually, the information is only transmitted at the speed of sound in the string or rod.

"Parts per million" isn't a good measure here. I assume you mean each "100 nanometer" piece to be a "part" of gold but what is a "part" of water? You would be better of calculating density in terms of grams of gold per gram of water.

What is preventing you from making one yourself? It couldn't take more than about 30 minutes.

If you are in empty space, nothing around. Just a waste space of nothingness. You could be traveling at the speed of light, casue you don't have anything relative to you. How would you know?

 

If 1000 people without the knowledge of time or clocks ended up on, say a space station without anything relative to them, how would time look like to them? How would time evolve?

Do you at least understand that "speed" has to be measured relative to something? Your first question is simply meaningless.

As for your second question, there are still things such as heartbeat, breathing rate, etc. that give a rough measure of time.