A scientist was carrying out an experiment on electricity. Suddenly his apparatus blew up. When he realized what had gone wrong, he was horrified at what he had accidentally done. He couldn't believe he had just touched the anode of his electrolytic cell.
He was positively shocked.
Batches of 15 steel balls are given to you. 14 of them are exactly alike. One ball is slightly heavier in each batch. You have a two pan balance that you can use to weigh the balls against each other. As the plant manager you are given the job of designing a method to find the heavier ball in each batch. Each weighing of any group of balls is considered one move. What is the average number of moves to determine the heavier ball for each batch?
What is the proof for the two common definitions of e? The continuous compounding and the sum over inverse factorials?
Definitions are not supposed be proved. It makes no sense to "prove" a definition. I think what you're looking for is a proof of the equivalence of two definitions of e.
I still don't understand why the bird has to accelerate. Can't it just fly in such a way that its forward speed is the same as the train's so it can just enter the train by slipping in sideways? Its overall velocity will of course be greater than 50mph, but the forward component of its velocity is 50mph. The train passengers will see it move sideways, but it will be stationary to them in the forwardsbackwards direction.
But even before you land, your motion will have a small additional velocity over the trains. Otherwise you would not have entered the train. Relative to the train once in the car you are travelling forward. That is why you roll forward.
What if you don't land? What if you can fly like the bird and you enter the train without making contact with any part of it?
I have placed the link you want in post #3.
By the way, I wished you have told me his name is Melonhead.
When I try to post a link, I get this message:
Sorry. In an effort to stop automated spam only established members can post links. Please describe where instead.
BTW, neither the poster nor the poster's friend is my friend. They're just members of that forum (of which I'm also a member). My conclusion is that the bird is stationary with respect to the train passengers, but the poster (Melonhead) doesn't agree, claiming that it's counterintuitive to see a bird flapping its wings like mad and still going nowhere.
This is a problem proposed by someone from another forum:
I have been having an ongoing disagreement with a friend about the outcome of a hypothetical situation involving a train and a bird. I'm hoping someone from this forum will be able to help me understand the physical laws that support my argument OR shoot me down in flames and tell me where Ive gone wrong.
Let's imagine that there is a train travelling North at a constant speed of 50mph. Outside, there is a bird flying parallel with the train that is also moving North at a constant speed of 50mph. (with me so far?). The bird then edges closer to the train and while still facing North the bird enters the train via a window.
I propose that once inside the train, assuming the bird continues to flap at its constant rate; it will fly towards the front of the train. Someone inside the train will observe the bird moving forward through the train at 50mph.
My friend proposes that the bird will stay at the same point in the train that it entered. I.e. if it entered at the back of coach E, even though its still flapping like mad, it will remain at the back of coach E and will appear stationary to an observer within the train.
What worries me is how blindingly obvious it seems to me that Im right. This feeling often coincides with me being wrong.
So who is right, the poster or their friend? I made some replies on that forum but it seems they were not being taken kindly to. I'd like to link you to the original discussion in the other forum but it seems I can't post links on this forum.
When you pass from a statement or set of statements to a new statement by logical deduction, you are said to infer the new statement. Example:
(i) All dogs have four legs.
(ii) Rover is a dog.
(ii) Therefore Rover has four legs.
The last statement is an inference from the first two.
(1) ifthen the system will not always be consistent (if consistent it will have infinitely many solutions),
(2) ifthen it will be consistent (and its solution is unique),
(3) ifthen it will also be consistent (but will have infinitely many solutions).
Addto the first equation and to the second equation. By using a combination of the commutative and associative axioms for addition, you should get
By uniqueness,as required.
Now following theorem holds: There is precisely one element (which will be denoted ), which is solution of equation .
Proof: Assume some fixed . Let the only solution of be denoted by symbol . So holds.
Now proof is straightforward and I will not finish it.