Three assertions, Which are true? Which are false? Why? Options

[*mipadi*]

I'm going to make three assertions. Which of these are true, which are false, and why?

Assertion 1

Assume you have a cute little bunny rabbit, and you place her thirty feet away from a carrot. The bunny really wants the carrot, but she can only get to it by jumping halfway to the carry each time she hops. Clearly the poor bunny will never make it to the carrot, because the distance between her and the carrot can be divided in half an infinite number of times (i.e. no matter where she is, the distance left can always be divided in two).

Now assume that the bunny can move however she wants. She still will never reach the carrot, because to do so, she must move through the same points as in the previous example; but as noted, there are an infinite number of points to move through, which means the bunny can never, ever reach that carrot. In fact, to move at all, the bunny must move through an infinite number of points; so unfortunately, the bunny is rooted to whatever spot she starts out from. Poor bunny.

Assertion 2

I'm mad at you, and I throw a rock at your head. You run away and try to avoid the rock. But why even bother? Assume I throw the rock at you. Now, take note of the position of that rock at a single instance of time. It's not moving, is it? At a specific moment in time, the rock has a position, but it does not have time to move and so is at rest. During each instance of time, it is at rest for the same reason. Therefore, the rock is always at rest, and it never moves. Therefore, it is foolish for you to run away.

Assertion 3

Assume a tortoise and a sprinter race. The sprinter is a nice guy and knows he can beat the tortoise, so he lets the tortoise start a hundred feet ahead of him. The sprinter and tortoise then both begin the race and, upon reaching their maximum speed, continue running at that speed. Clearly the tortoise is much, much slower than the sprinter—but will the sprinter ever catch up to the tortoise? Unfortunately, the answer is no, he won't.

Assume the sprinter has run a hundred feet, and thus reached the tortoise's starting point. In this time, the tortoise has moved a foot, which still means he is beating the sprinter by a foot. The sprinter than runs another foot, catching up to the tortoise's position—or so it seems, but remember that the tortoise has moved during this period of time, too, so he is still a tiny bit ahead of the sprinter. The sprinter takes a tiny bit of time to reach this point, but in this period of time, the tortoise has moved forward a tiny bit more again. Thus, whenever the sprinter reaches a point where the tortoise has been, the tortoise has always moved a bit farther forward. No matter how fast the sprinter runs, the tortoise is always ahead. Wasn't he foolish for giving that tortoise a head start?!

[BOOGERSHAHA]

no real opinion, but i was wondering where the last situation came from. i think i've read something similar to it in a book before...

[*mipadi*]

no real opinion, but i was wondering where the last situation came from. i think i've read something similar to it in a book before...

All three deal with a specific application of physical or mathematical knowledge, but they are largely contrived situations to highlight that application. The concepts dealt with in each assertion are likely to be found elsewhere, as they mostly apply to general knowledge in mathematics and natural sciences.

[*RiC3xBoy*]

I am a bit confused by the 2nd one. It's prob just me, but I don't really understand. For assertions 1 and 3, you can argue both sides to be true or false, I'm going to say true for the first one and false to the 3rd.

For 1: Although the bunny is moving, he will continue to get infinitely closer to the carrot, but he will never reach it.

For 3: The sprinter is going at a faster constant velocity than the turtoise, so unless the question is asking within a limited time, the sprinter will pass the turtoise.

Also, Yes, I know I am pretty much contradicting myself, but my main message was that, if you were to look at the assertions in different perspectives, you would get different answers.

[*kryogenix*]

I guess for the first one, the graph of the function of the bunny's position will be asymptotic, so it will never intersect with the position of the carrot for the first part. But for the second part, what's stopping the rabbit from going from 0-30? The rabbit will hit the infinite value of points between 0-30, but will still make it to the carrot. I guess it's like the mean value theorem, if I remember correctly

I think assertion 2 is false. Even though the rock's position is static for that moment in time, it still has a positive acceleration.

Assertion 3 is false. If you graph both the sprinter and the tortoise, the lines will intersect eventually, since the sprinter has a greater speed and therefore, a steeper slope.

False for all I guess.

[*wind&fire*]

false for the first one
where the rabbits' movement are classified as an "infinite geometric series"
e.g.
therefore to get to the carrot distances to get there would be
15+7.5+3.75+1.875...
this represents a infintie geometric series... there is a limiting sum as -1<r<1

S(initfinity) = times they have to jump

a = first jump = 15

r= ratio .5

S(initfinity) = a/(1-r)
= 15/(1-.5)
= 30
thus the rabbit must jump 30 times to reach the goal...(i hope its right... i havent done maths in 6 months)
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second one i think is false too as if we take into consideration the position of the rock being stationary therefore we must be stationary.. youve stopped time therefore were not moving either thus none of shall ever meet... if you examine the situation in instances then i would still be able dodge the rock since projectiles follow a predicatable parabolic path thus i would be able to shift as to miss the rock...
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im not sure wha the last one is since the end of the race was never specified and what constitutes as "race" ... here you assume being behind someone in a race is losing and were running in a loop...

This post has been edited by wind&fire: Mar 10 2006, 08:10 AM

[*salcha*]

I'm assuming Assertion 1 is true, because of the infinite number of times you can divide. Unless you state a limit, one will never get to the carrot. Actually, I'm actually thinking about the second situation....

Uhhh, assertion 3 is false. The sprinter is always running at a faster length and speed than the tortoise. If the sprinter was able to catch up to the 100 feet that the tortoise had moved and it had taken the tortoise to only move one foot in that time length, the sprinter will of course pass the tortoise eventually. You didn't state how long the race was, however..

[racoons > you]

point 1 is technically true, because of infinitely dividing thing sin half,and whatnot. however, eventually, it wil lbecome irrelevant, as the distance will become so small that the rabbit can reach to bite into the carrot without needing to hop.

point 2 is false, because as james says, the rock has positive acceleration. furthermore, althouh it is stationary at each individual moment in time, time itself moves, giving motion to the rock, in manner of a realyl detailed flip book, if you will.

i dont understand point three, but the sprinter will reach the staring line ahead of the tortoise, in any case.

[*mipadi*]

Whoops, almost forgot about this.

All the assertions are false. But why? A lot of good points were made (the fellow who brought up the infinite geometric series was pretty much on the right track!). Here's a few hints: In the first one, think about time. Look at the time it takes to reach the carrot, not the distance. Thinking in terms of time helps with the third assertion, as well.

kryogenix was pretty much on the right track for the second one, but think of the instantaneous velocity, not acceleration, at a given point. (Derivatives!)

Good thinking. Sorry to all that I took so long to respond. Hope you enjoyed the fun problems.

[IceCream4U]

^Man, I just typed a long response. Then I saw that post..