[Guest]

A ant starts at the beginning of a straight rubber band (not a loop) 10 inches long and moves at a speed of 1 inch/sec. After the ant starts its journey, the rubber band is stretched one inch every second. Assume that the placement of the ant is wherever its front legs are.

How long does it take the ant to reach the end and how long will the rubber band be when it does?

[Guest]

Been done - se prof Templeton's topics - i think it was

[Guest]

A ant starts at the beginning of a straight rubber band (not a loop) 10 inches long and moves at a speed of 1 inch/sec. After the ant starts its journey, the rubber band is stretched one inch every second. Assume that the placement of the ant is wherever its front legs are.

How long does it take the ant to reach the end and how long will the rubber band be when it does?

the ant's position on the band gets 'stretched' along with the rest of the band, his effective velocity is increased.

Therefore the ant will still take just 10 seconds to walk the length of the band, at which point it will have doubled in length to 20 inches.

[Guest]

The post by Prof Templeton differed slightly. It involved the rubber band stretching constantly and at a larger rate.

The speed of the ant is constant at 1 inch/sec. It is not relative to the rubber band. Thus, the stretching doesn't not affect the ant's velocity.

Edited January 6, 2009 by vinays84

[Guest]

Well! If the velocity of ant is independant of stretch of the band then I guess

the distance between the ant and the end of the band would remain the same, until the band cannot be stretched anymore. (Ofcourse, the band cannot be stretched infinitely).

[Guest]

A ant starts at the beginning of a straight rubber band (not a loop) 10 inches long and moves at a speed of 1 inch/sec. After the ant starts its journey, the rubber band is stretched one inch every second. Assume that the placement of the ant is wherever its front legs are.

How long does it take the ant to reach the end and how long will the rubber band be when it does?

What will the maximum length of the rubber band be?

[bonanova]

The speed of the ant is constant at 1 inch/sec. It is not relative to the rubber band. Thus, the stretching doesn't not affect the ant's velocity.

If the ant does move relative to the band, then:

1. The ant starts out 10" from the far end of the rubber band.
   
2. Both the ant and the far end move forward at [whatever speed, they are equal].
   
3. The ant never gets to the end - their separation is forever 10"
   

Seriously though,

How can the ant's feet be attached to the rubber band and its motion be relative to something other than the rubber band?

Is the ant flying at 1"/second?

Is the ant, instead, walking on the ground, next to the rubber band?

Or was that a typo?

[Prof. Templeton]

The band gets to be 27" long.

[Guest]

Perhaps my wording was a little cryptic, so I'll try to clear some stuff up:

The ant moves along the rubber band at a constant 1 inch/sec. When the rubber band is stretched, the entire rubber band is stretched, not just the end, and while it will affect the ant's position on the rubber band, it will not affect its velocity. The rubber band also can be stretched to infinite length.

How can the ant's feet be attached to the rubber band and its motion be relative to something other than the rubber band?

The ant's motion is relative to the rubber band as well as the floor below. The point I wanted to make is that when the rubber band is stretched, the ant will not gain any velocity from the stretching. It might be easier to imagine a tiny person walking the rubber band instead, so that there is only one point of contact (no back legs).

Edited January 7, 2009 by vinays84

[Guest]

Just a guess...

If the rubber band is increasing at a rate of 1"/s, then it is [probably] safe to assume that each end is increasing 0.5"/s. If this is true, then in the case of the ant traveling at 1"/s, we can say that

1t=10+0.5t where t is time. solving this we have the time the ant takes to get to the end of the rubber band as being 20 seconds. If the rubber band is increasing in length of

L=10+1t, and t is 20 seconds, then the length of the rummber band would be 30".

...??

[Guest]

This puzzle seems to be a trick. The band's final length isn't specified (i.e. it could stretch to infinity), and if the band is stretching at the same velocity as the ant is walking then it would take infinite time to reach the end.

I could be way off base here, but I think we need more parameters to give a finite time.

[Guest]

wont the band eventually break and the ant will fly across the room smack into something and die?

[Guest]

The ant moves along the rubber band at a constant 1 inch/sec. When the rubber band is stretched, the entire rubber band is stretched, not just the end, and while it will affect the ant's position on the rubber band, it will not affect its velocity. The rubber band also can be stretched to infinite length.

The ant's motion is relative to the rubber band as well as the floor below. The point I wanted to make is that when the rubber band is stretched, the ant will not gain any velocity from the stretching. It might be easier to imagine a tiny person walking the rubber band instead, so that there is only one point of contact (no back legs).

You don't specify it the band is being stretched from both ends, or from one end - presumably the distal end, away from senor ant.

If you pull from both ends, it seems that there would be a point in the middle of the band which stays still relative to the ground. each end moves away from this middle point at 1/2 inch/sec as was pointed out.

If you instead anchor the end where the ant starts and then pull towards infinity from the other end, all points move away from this static point.

In either case, since the band is moving relative to the surface, the ant's velocity cannot be the same relative to the surface and relative to the band. My original answer assumed that the ant was in the reference frame of the rubber band, since its feet were pushing off of this moving surface their velocities would be additive. I had further assumed the second stretching mode - if you assume the first it seems logical that until the ant reaches the midpoint the movement of the band would have a net decelerating effect on its motion relative to the surface, and after it had reached the midpoint it would have a net accelerating effect.

[Guest]

The rubber band is anchored at one end and stretched from the end that the ant is trying to reach. Like I've mentioned three times already, the ant does NOT gain extra velocity when the rubber band is stretched. It has a CONSTANT speed of 1/inch per second.

In either case, since the band is moving relative to the surface, the ant's velocity cannot be the same relative to the surface and relative to the band. My original answer assumed that the ant was in the reference frame of the rubber band, since its feet were pushing off of this moving surface their velocities would be additive. I had further assumed the second stretching mode - if you assume the first it seems logical that until the ant reaches the midpoint the movement of the band would have a net decelerating effect on its motion relative to the surface, and after it had reached the midpoint it would have a net accelerating effect.

The band is not moving relative to the surface. It is stretched an inch every second, not constantly, as stated in the original post.

[Guest]

The rubber band is anchored at one end and stretched from the end that the ant is trying to reach. Like I've mentioned three times already, the ant does NOT gain extra velocity when the rubber band is stretched. It has a CONSTANT speed of 1/inch per second.

The band is not moving relative to the surface. It is stretched an inch every second, not constantly, as stated in the original post.

IMHO, the wording in the original post was ambiguous as to whether it was stretched incrementally or continuously.

Any arbitrary point in the rubber band does move relative to the surface when it is stretched, except the anchor point. The point at the distal end moves 1 inch with each stretching - unless you are adding rubber at the far end, instead of stretching.

If an ant is standing on the band when it is stretched, it is going to move relative to the surface.

[Guest]

IMHO, the wording in the original post was ambiguous as to whether it was stretched incrementally or continuously.

Any arbitrary point in the rubber band does move relative to the surface when it is stretched, except the anchor point. The point at the distal end moves 1 inch with each stretching - unless you are adding rubber at the far end, instead of stretching.

If an ant is standing on the band when it is stretched, it is going to move relative to the surface.

You are correct in all you've said, including the wording. I should have said "stretched every second, on the second". So, if you know that the ant will move relative to the surface with each stretch, but will maintain its velocity of 1/inch per second, can you find the answer?

[Guest]

I think people are still relating this problem to the previous solution, and not looking at the differences in this puzzle. The important thing to note about the stretching here is that it is instantaneous. At time just under 1 second, the ant has traveled almost 1 inch on a 10 inch band. At time just over 1 second, the ant has traveled 1.1 inches on an 11 inch band. In this case the majority of the elongation happened in front of the ant, but not the entire inch. The proportion of the stretching that happens in front of and behind the ant changes as he moves along the band.

I go tthe same answer as the Proffessor did using this method.

27 inches long, at about 17.7 seconds

[Guest]

I think people are still relating this problem to the previous solution, and not looking at the differences in this puzzle. The important thing to note about the stretching here is that it is instantaneous. At time just under 1 second, the ant has traveled almost 1 inch on a 10 inch band. At time just over 1 second, the ant has traveled 1.1 inches on an 11 inch band. In this case the majority of the elongation happened in front of the ant, but not the entire inch. The proportion of the stretching that happens in front of and behind the ant changes as he moves along the band.

I go tthe same answer as the Proffessor did using this method.

27 inches long, at about 17.7 seconds

If you assume the stretching happens every second on the second starting at 1 second (after the ant has marched one inch) I get:

15.33 seconds, at which time the ant has marched 24 inches. (which, by the way gives it an average velocity of 1.57"/second relative to the surface!)

If the band starts stretching at time 0, it takes the ant 18.75 seconds and he will have traveled 29.25 inches.

[Prof. Templeton]

If you assume the stretching happens every second on the second starting at 1 second (after the ant has marched one inch) I get:

15.33 seconds, at which time the ant has marched 24 inches. (which, by the way gives it an average velocity of 1.57"/second relative to the surface!)

If the band starts stretching at time 0, it takes the ant 18.75 seconds and he will have traveled 29.25 inches.

The OP was a little vauge on that but I think me are meant to start the stretch at t=0. IMO

[Guest]

The OP was a little vauge on that but I think me are meant to start the stretch at t=0. IMO

I also think so and found the same answer with Prof, mike and ~xucam.

But I got it by dummy method (software at pc).

Otherwise I should have used calculus, though I tried it and messed it up.

I wonder how you guys got the result?

[Guest]

I meant that the band would be stretched at the first second and then every second following. If it were stretched at 0 seconds, I might as well have said that the original length of the rubber band was 11.

Anyway, I also arrived at a

If the band is stretched starting at second 1, the ant takes 16.338 seconds and the rubber band would therefore be 26 inches long.

If the band is stretched starting at second 0, the ant takes 18.052 seconds and the rubber band would therefore be 29 inches long.

[Prof. Templeton]

I meant that the band would be stretched at the first second and then every second following. If it were stretched at 0 seconds, I might as well have said that the original length of the rubber band was 11.

Anyway, I also arrived at a

If the band is stretched starting at second 1, the ant takes 16.338 seconds and the rubber band would therefore be 26 inches long.

If the band is stretched starting at second 0, the ant takes 18.052 seconds and the rubber band would therefore be 29 inches long.

What I meant by starting at t=0 is that is when the stretch starts, so at t=1 the band is 11 inches (it stretched from 10 to 11 in that first second, but it began to stretch as soon as we left t=0).

What did you use for "e", because I can get different answers depending on if a constant is used or a faction.

[Guest]

A ant starts at the beginning of a straight rubber band (not a loop) 10 inches long and moves at a speed of 1 inch/sec. After the ant starts its journey, the rubber band is stretched one inch every second. Assume that the placement of the ant is wherever its front legs are.

How long does it take the ant to reach the end and how long will the rubber band be when it does?

At the end of 1 second ant has moved 1 inch and rubberband has been stretched 1 inch but 1/10th of the stretched rubberband has already been covered by the ant, or ant has actually covered 1.10 inches, and will have 9.9 inches remaining to go to the end. By this logic ant will have to cover 100 inches.

[Guest]

What I meant by starting at t=0 is that is when the stretch starts, so at t=1 the band is 11 inches (it stretched from 10 to 11 in that first second, but it began to stretch as soon as we left t=0).

What did you use for "e", because I can get different answers depending on if a constant is used or a faction.

What do you mean by "e" and faction? Are you solving this by hand mathematically? If so, I'd love to see how you do it. I started to do it mathematically and was able to come up with formulas to find the position at any second, but they were in summations and I didn't know how to change them into expressions I could easily evaluate.

I had reached my answers using excel.

[Prof. Templeton]

What do you mean by "e" and faction? Are you solving this by hand mathematically? If so, I'd love to see how you do it. I started to do it mathematically and was able to come up with formulas to find the position at any second, but they were in summations and I didn't know how to change them into expressions I could easily evaluate.

I had reached my answers using excel.

If you plot out your excel points they should form a hyperbolic curve (or a portion of). In order to solve for the area under that curve you need to use natural logs, which are done in base e, so that's where the e comes in. Otherwise your just added up "steps" under (or centered on) the curve.

[Guest]

If you plot out your excel points they should form a hyperbolic curve (or a portion of). In order to solve for the area under that curve you need to use natural logs, which are done in base e, so that's where the e comes in. Otherwise your just added up "steps" under (or centered on) the curve.

As I had the values in excel, I simply added them in excel (no mathematics), which is how I arrived at my answer.

I did come up with the formula that d(n) = 1 + Σ(x=0->n-1) of (1/(10+x)) where d(n) is the distance traveled by the ant during the nth second (including the distance he gains from the nth stretch).

I couldn't figure out how this could be translated, however.