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The traveling ant

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An old favorite.

Imagine an infinitely stretchy elastic band is tied to a tree on one end and to the bumper of a pickup truck on the other end. The truck, with an infinite tank of gas, drives away from the tree on an infinitely long straight road at a constant speed of 10 mph. Andy Ant walks with a constant speed of 0.1 mph relative to whatever surface he is standing on. When the truck has reached a point 0.1 miles from the tree, you place Andy on the elastic band at the tree and start him walking toward the truck.

Does Andy ever reach the truck?

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Posted (edited) · Report post

Define the ant's rate of travel as its constant speed plus the speed of the band where it stands.

Constant speed = 0.1

Speed of band is the ratio of the ant's current location (x) to the location of the truck times 10mph (assuming the band's stretch is consistent from start to finish). The current location of the truck is 10t, so the speed of the band is x/t.

dx/dt=0.1+x/t

Solving the ODE gives x=t(c+0.1*ln(t))

Given x(.01)=0, (from: truck going 10mph, hits 0.1m at 0.01h)

0=.01*(c+.1ln(.01)), c=-0.1ln(0.01)

Finding intersection:

0=t(-0.1ln(0.01)+0.1*ln(t))-10t

gives, according to wolfram alpha,

x~~-0.000320538-0.000248259 i

Which, unless something went wrong, means poor Andy will never reach that truck.

Edited by jamieg
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Posted · Report post

I suppose Andy is thought of as a creature of no dimensions (a moving point), and that the rope is already fully stretched (not slacking) when Andy starts walking.

Let us define a unit called band length (bl), as always being the length of the band in its current state. Of course this unit will grow larger over time. The truck is always 1 bl away from the tree. When Andy is 1 bl away from the tree, he will have reached the truck.

Given a point in time (x hours), let's say Andy is y bl away from the tree. If Andy were to stop walking at this point, he would forever remain y bl away from the tree. He would be moving further from the tree in terms of miles, but not in terms of the ever-increasing unit band length. So the part of his movement which is caused by the rope stretching always equals 0 band lengths per hour (blph). The only part of his movement that counts in terms of band length is his walking speed.

So, at the time x hours, the band length is 0.1+10x miles. Andy is walking with a speed of 0.1 mph, which equals 0.1/(0.1+10x) blph. Hence, the distance Andy has traveled in band lengths is integralt=0x 0.1/(0.1+10t)dt = [0.01*ln(0.1+10t)]0x = 0.01*(ln(0.1+10x)-ln(0.1)) = 0.01*ln((0.1+10x)/0.1) = 0.01*ln(1+100x). Now all we need to do is solve the equation 0.01*ln(1+100x) = 1.

ln(1+100x) = 100

1+100x = e^100

100x = e^100+1

x = (e^100+1)/100 ~= 2.69*10^41 hours.

Answer: yes, but only if he is immortal or has an extremely long life-span.

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Posted · Report post

This reminds me of a scooter trip I once took on these very boards where I learned a valuable lesson about head-starts.

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Posted (edited) · Report post

the answer is yes. Andy the ant is placed on the elastic band at the point of 1/100 an hour, or after 36 seconds.

as noted by rainman, (though i don't agree with his result) if the ant stands still the relative distance to the truck will remain the same.

so, after the next 36 seconds, the elastic band will double, and andy will be at slightly more than 1/10000 of the way. after the next 36 seconds, the elastic band will be 3/2 larger, putting andy at roughly 3/20000 +1/10000 = 5/20000. after the next 36 seconds, the elastic band will be 4/3 more, putting andy at roughly 20/60000 +1/10000 = 26/60000.

more precisely, we have... e^(0.01*x) -1 = 100; or that andy will reach the truck after 461.5 hours roughly.

Edited by phil1882
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Posted (edited) · Report post

the answer is yes. Andy the ant is placed on the elastic band at the point of 1/100 an hour, or after 36 seconds.

as noted by rainman, (though i don't agree with his result) if the ant stands still the relative distance to the truck will remain the same.

so, after the next 36 seconds, the elastic band will double, and andy will be at slightly more than 1/10000 of the way. after the next 36 seconds, the elastic band will be 3/2 larger, putting andy at roughly 3/20000 +1/10000 = 5/20000. after the next 36 seconds, the elastic band will be 4/3 more, putting andy at roughly 20/60000 +1/10000 = 26/60000.

more precisely, we have... e^(0.01*x) -1 = 100; or that andy will reach the truck after 461.5 hours roughly.

I think your first sentence is wrong

If the ant was to stand still, let's say exactly in the middle of the band, it will always be in the middle of the band, if it stood at 1 quarter of the length of the band, it will always be at one quarter of the length of the band and so on...

You can imagine splitting the band to N points and marking them, now if the ant stood between any two points when the band stretches it will always be between those two points

But you can put it another way:

get a marker and mark N points on the band at equal distances, the band grows linearly, it's length at time t would be

L(t) = L0 + v0t, you can see that the distance between every two points would be exactly 1/N * L(t), take the dirivative and you'll see that the growth rate of that piece is v0/N.

So if we take an N that is big enough so that v0/N is less than the ant's walking speed then the ant could get from mark k to mark k+1 in finite time, and since there is a finite number of points it will get to the last point in finite time as well,,,

Edited by Anza Power
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Posted · Report post

the answer is yes. Andy the ant is placed on the elastic band at the point of 1/100 an hour, or after 36 seconds.

as noted by rainman, (though i don't agree with his result) if the ant stands still the relative distance to the truck will remain the same.

so, after the next 36 seconds, the elastic band will double, and andy will be at slightly more than 1/10000 of the way. after the next 36 seconds, the elastic band will be 3/2 larger, putting andy at roughly 3/20000 +1/10000 = 5/20000. after the next 36 seconds, the elastic band will be 4/3 more, putting andy at roughly 20/60000 +1/10000 = 26/60000.

more precisely, we have... e^(0.01*x) -1 = 100; or that andy will reach the truck after 461.5 hours roughly.

Your answer lacks explanations.

"so, after the next 36 seconds, the elastic band will double, and andy will be at slightly more than 1/10000 of the way." I don't see how the part in bold follows from anything you stated before that. I also don't see how it can be true. At the moment Andy starts walking, the band is 0.1 miles long and Andy is walking at 0.1 miles per hour. At this moment his speed would take him all the way in one hour, or 1/100 of the way in 36 seconds. 36 seconds later, when your first time interval ends, the band is 0.2 miles long and Andy is still walking at 0.1 miles per hour. At this moment his speed would take him all the way in two hours, or 1/200 of the way in 36 seconds. So in a time interval where his walking speed goes from 1/100 band lengths per 36 seconds to 1/200 band lengths per 36 seconds, how did he only manage to walk slightly more than 1/10000 band lengths in those 36 seconds?

"more precisely, we have... e^(0.01*x) -1 = 100..." This also seems taken right out of the blue, no explanation given to how you came up with this formula.

"as noted by rainman, (though i don't agree with his result)..." It is not good practice to disagree with a result unless you disagree with something which leads to said result. My result might be wrong, but only if some part of my solution is wrong. A false result can not be reached by a correct solution. So rather than simply saying you disagree with my result, please explain that statement by pointing to a fault in my solution.

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