A car is moving at 40 km/h. A fly at 100 km/h, starts from wall towards the car(20 km away)flies to car and...












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A car is moving at a constant speed 40 km/h along a straight road which heads towards a wall.A fly flying at a constant speed of 100 km/h, starts from wall the towards the car at a instant when the car is 20 km away, flies until it reaches the car and comes back to the wall at the same speed.It continues to fly between the car and the wall till the car reach the wall. How many trips has it made between the car and the wall?



I don't expect a brute force because I already did that. Some thing like arithmetic/geometric/harmonic progression will satisfy my curiosity.










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$endgroup$








  • 6




    $begingroup$
    In theory infinitely many.
    $endgroup$
    – fleablood
    11 hours ago






  • 4




    $begingroup$
    This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
    $endgroup$
    – RedGrittyBrick
    9 hours ago








  • 2




    $begingroup$
    Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
    $endgroup$
    – KJO
    8 hours ago






  • 2




    $begingroup$
    At least it's not a spherical cow....
    $endgroup$
    – Kathy
    6 hours ago






  • 1




    $begingroup$
    Possible duplicate of 'Bee flying between two trains' problem
    $endgroup$
    – Herohtar
    5 hours ago
















4












$begingroup$


A car is moving at a constant speed 40 km/h along a straight road which heads towards a wall.A fly flying at a constant speed of 100 km/h, starts from wall the towards the car at a instant when the car is 20 km away, flies until it reaches the car and comes back to the wall at the same speed.It continues to fly between the car and the wall till the car reach the wall. How many trips has it made between the car and the wall?



I don't expect a brute force because I already did that. Some thing like arithmetic/geometric/harmonic progression will satisfy my curiosity.










share|cite|improve this question









$endgroup$








  • 6




    $begingroup$
    In theory infinitely many.
    $endgroup$
    – fleablood
    11 hours ago






  • 4




    $begingroup$
    This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
    $endgroup$
    – RedGrittyBrick
    9 hours ago








  • 2




    $begingroup$
    Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
    $endgroup$
    – KJO
    8 hours ago






  • 2




    $begingroup$
    At least it's not a spherical cow....
    $endgroup$
    – Kathy
    6 hours ago






  • 1




    $begingroup$
    Possible duplicate of 'Bee flying between two trains' problem
    $endgroup$
    – Herohtar
    5 hours ago














4












4








4


1



$begingroup$


A car is moving at a constant speed 40 km/h along a straight road which heads towards a wall.A fly flying at a constant speed of 100 km/h, starts from wall the towards the car at a instant when the car is 20 km away, flies until it reaches the car and comes back to the wall at the same speed.It continues to fly between the car and the wall till the car reach the wall. How many trips has it made between the car and the wall?



I don't expect a brute force because I already did that. Some thing like arithmetic/geometric/harmonic progression will satisfy my curiosity.










share|cite|improve this question









$endgroup$




A car is moving at a constant speed 40 km/h along a straight road which heads towards a wall.A fly flying at a constant speed of 100 km/h, starts from wall the towards the car at a instant when the car is 20 km away, flies until it reaches the car and comes back to the wall at the same speed.It continues to fly between the car and the wall till the car reach the wall. How many trips has it made between the car and the wall?



I don't expect a brute force because I already did that. Some thing like arithmetic/geometric/harmonic progression will satisfy my curiosity.







algebra-precalculus






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 11 hours ago









user654700user654700

334




334








  • 6




    $begingroup$
    In theory infinitely many.
    $endgroup$
    – fleablood
    11 hours ago






  • 4




    $begingroup$
    This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
    $endgroup$
    – RedGrittyBrick
    9 hours ago








  • 2




    $begingroup$
    Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
    $endgroup$
    – KJO
    8 hours ago






  • 2




    $begingroup$
    At least it's not a spherical cow....
    $endgroup$
    – Kathy
    6 hours ago






  • 1




    $begingroup$
    Possible duplicate of 'Bee flying between two trains' problem
    $endgroup$
    – Herohtar
    5 hours ago














  • 6




    $begingroup$
    In theory infinitely many.
    $endgroup$
    – fleablood
    11 hours ago






  • 4




    $begingroup$
    This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
    $endgroup$
    – RedGrittyBrick
    9 hours ago








  • 2




    $begingroup$
    Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
    $endgroup$
    – KJO
    8 hours ago






  • 2




    $begingroup$
    At least it's not a spherical cow....
    $endgroup$
    – Kathy
    6 hours ago






  • 1




    $begingroup$
    Possible duplicate of 'Bee flying between two trains' problem
    $endgroup$
    – Herohtar
    5 hours ago








6




6




$begingroup$
In theory infinitely many.
$endgroup$
– fleablood
11 hours ago




$begingroup$
In theory infinitely many.
$endgroup$
– fleablood
11 hours ago




4




4




$begingroup$
This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
$endgroup$
– RedGrittyBrick
9 hours ago






$begingroup$
This 100 km/h fly is a very impressive aviator given that ordinary houseflies have a max speed of 8 km/h. Even more impressive is its ability to instantaneously go from +100 km/h to -100km/h without feeling a bit groggy for a moment.
$endgroup$
– RedGrittyBrick
9 hours ago






2




2




$begingroup$
Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
$endgroup$
– KJO
8 hours ago




$begingroup$
Everyone seems to forget Applied Mathematic is not as it should be when the car reaches the point where the gap is less than the size of the flies body it dies of the pressure if not exhaustion.
$endgroup$
– KJO
8 hours ago




2




2




$begingroup$
At least it's not a spherical cow....
$endgroup$
– Kathy
6 hours ago




$begingroup$
At least it's not a spherical cow....
$endgroup$
– Kathy
6 hours ago




1




1




$begingroup$
Possible duplicate of 'Bee flying between two trains' problem
$endgroup$
– Herohtar
5 hours ago




$begingroup$
Possible duplicate of 'Bee flying between two trains' problem
$endgroup$
– Herohtar
5 hours ago










4 Answers
4






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Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.



The fly and and car have a combined speed of $140 frac {km}{hr}$ so the fly reaches the car in $frac h{140}$ hours. In that time the car has traveled $40frac h{140} = frac 27h$ and is now $h-frac 27h = frac 57h$ from the wall. So the fly heads back to the wall.



As the trip back is just as far this takes $frac h{140}$ hours and the car has traveled another $frac 27h$ and is now $frac 37h$ from the wall. [1]



So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.



Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.



If one wishes to set this up as an infinite sum.....



Each trip the fly flies $frac {10}7$ of the distance the car was away. And each trip the car is $frac {3}{7}$ of the distance it was before. So the distance the fly travels is $sum_{k=0}^{infty} frac {10}7*(frac {3}{7})^k*20$ which if I did this correctly is



$frac {10}7*20(sum_{k=0}^{infty} (frac {3}{7})^k)= frac {200}{7}frac 1{1-frac {3}{7}} = frac {200}{7}frac {7}{4}= 50$km.



[1](for the record, in this time, $frac 1{70}h$ hours, the car has traveled $frac 47h$ and the fly has travelled $frac {10}7h$).






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$endgroup$





















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    $begingroup$

    It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.



    The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      I got that but can you please give me more rigorous proof? If you don't mind.
      $endgroup$
      – user654700
      11 hours ago










    • $begingroup$
      @user654700 What exactly do you find not rigorous enough?
      $endgroup$
      – Arthur
      11 hours ago






    • 8




      $begingroup$
      @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
      $endgroup$
      – Ross Millikan
      11 hours ago












    • $begingroup$
      I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
      $endgroup$
      – inavda
      11 hours ago










    • $begingroup$
      Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
      $endgroup$
      – fleablood
      11 hours ago



















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    Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.



    Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.






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    • $begingroup$
      But how many trips is this?
      $endgroup$
      – Kathy
      6 hours ago










    • $begingroup$
      Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
      $endgroup$
      – Guest3711
      6 hours ago








    • 2




      $begingroup$
      I guess what I'm saying is you should edit your answer to include this.
      $endgroup$
      – Kathy
      6 hours ago



















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    $begingroup$

    Arthur's answer is very good, but here is another (equally valid) way of visualizing the problem:



    Let us graph the positions of the wall, fly, and car over time. The wall doesn't move, so it is represented by a horizontal line. The car starts at some distance away from the wall but moves towards it at a constant speed until it hits the wall -- so we have a line that intersects the wall's line. Now for the interesting part:



    The fly's path starts off as a line with greater slope than the car's line, until it hits the wall's line. The fly's speed remains the same, but it is going in the opposite direction. So now the path continues as if the wall's line was a mirror and it was reflected. When the fly hits the car, the same thing happens -- the fly's path is reflected and it continues towards the wall again (one caveat: when it bounces off the car's line, the angles of incidence and reflection are not equal so it isn't behaving exactly as light would).



    So we can see that the fly's path continues bouncing up and down, always at the same or opposite slope.



    The final thing to notice to grasp the intuition is that this diagram we have constructed is self-similar. If we zoom in so that the second bounce with the wall is where the first bounce used to be, we have the same exact diagram as before. I won't prove this, but if you draw it out, you can see it intuitively. Essentially, this means that no matter how close the car gets to the wall, we can zoom in and see more bounces for the fly.






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      4 Answers
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      4 Answers
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      11












      $begingroup$

      Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.



      The fly and and car have a combined speed of $140 frac {km}{hr}$ so the fly reaches the car in $frac h{140}$ hours. In that time the car has traveled $40frac h{140} = frac 27h$ and is now $h-frac 27h = frac 57h$ from the wall. So the fly heads back to the wall.



      As the trip back is just as far this takes $frac h{140}$ hours and the car has traveled another $frac 27h$ and is now $frac 37h$ from the wall. [1]



      So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.



      Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.



      If one wishes to set this up as an infinite sum.....



      Each trip the fly flies $frac {10}7$ of the distance the car was away. And each trip the car is $frac {3}{7}$ of the distance it was before. So the distance the fly travels is $sum_{k=0}^{infty} frac {10}7*(frac {3}{7})^k*20$ which if I did this correctly is



      $frac {10}7*20(sum_{k=0}^{infty} (frac {3}{7})^k)= frac {200}{7}frac 1{1-frac {3}{7}} = frac {200}{7}frac {7}{4}= 50$km.



      [1](for the record, in this time, $frac 1{70}h$ hours, the car has traveled $frac 47h$ and the fly has travelled $frac {10}7h$).






      share|cite|improve this answer











      $endgroup$


















        11












        $begingroup$

        Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.



        The fly and and car have a combined speed of $140 frac {km}{hr}$ so the fly reaches the car in $frac h{140}$ hours. In that time the car has traveled $40frac h{140} = frac 27h$ and is now $h-frac 27h = frac 57h$ from the wall. So the fly heads back to the wall.



        As the trip back is just as far this takes $frac h{140}$ hours and the car has traveled another $frac 27h$ and is now $frac 37h$ from the wall. [1]



        So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.



        Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.



        If one wishes to set this up as an infinite sum.....



        Each trip the fly flies $frac {10}7$ of the distance the car was away. And each trip the car is $frac {3}{7}$ of the distance it was before. So the distance the fly travels is $sum_{k=0}^{infty} frac {10}7*(frac {3}{7})^k*20$ which if I did this correctly is



        $frac {10}7*20(sum_{k=0}^{infty} (frac {3}{7})^k)= frac {200}{7}frac 1{1-frac {3}{7}} = frac {200}{7}frac {7}{4}= 50$km.



        [1](for the record, in this time, $frac 1{70}h$ hours, the car has traveled $frac 47h$ and the fly has travelled $frac {10}7h$).






        share|cite|improve this answer











        $endgroup$
















          11












          11








          11





          $begingroup$

          Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.



          The fly and and car have a combined speed of $140 frac {km}{hr}$ so the fly reaches the car in $frac h{140}$ hours. In that time the car has traveled $40frac h{140} = frac 27h$ and is now $h-frac 27h = frac 57h$ from the wall. So the fly heads back to the wall.



          As the trip back is just as far this takes $frac h{140}$ hours and the car has traveled another $frac 27h$ and is now $frac 37h$ from the wall. [1]



          So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.



          Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.



          If one wishes to set this up as an infinite sum.....



          Each trip the fly flies $frac {10}7$ of the distance the car was away. And each trip the car is $frac {3}{7}$ of the distance it was before. So the distance the fly travels is $sum_{k=0}^{infty} frac {10}7*(frac {3}{7})^k*20$ which if I did this correctly is



          $frac {10}7*20(sum_{k=0}^{infty} (frac {3}{7})^k)= frac {200}{7}frac 1{1-frac {3}{7}} = frac {200}{7}frac {7}{4}= 50$km.



          [1](for the record, in this time, $frac 1{70}h$ hours, the car has traveled $frac 47h$ and the fly has travelled $frac {10}7h$).






          share|cite|improve this answer











          $endgroup$



          Suppose the fly is at the wall. And suppose this will be the last trip or partial trip of the fly. Suppose the car is $h$ miles away.



          The fly and and car have a combined speed of $140 frac {km}{hr}$ so the fly reaches the car in $frac h{140}$ hours. In that time the car has traveled $40frac h{140} = frac 27h$ and is now $h-frac 27h = frac 57h$ from the wall. So the fly heads back to the wall.



          As the trip back is just as far this takes $frac h{140}$ hours and the car has traveled another $frac 27h$ and is now $frac 37h$ from the wall. [1]



          So the fly starts another trip, contradicting that this was his last. So the fly never makes a last trip an instead there are an infinite number of trips.



          Figuring out how far the fly flies is a matter of noting the car is on a straight path and travels $20km$ at $40 kmh$ so this takes $30$ minutes. The fly no matter how many times (infinitely many) it zigs will travel at $100kmh$. So in $30$ minutes it flies $50 km$.



          If one wishes to set this up as an infinite sum.....



          Each trip the fly flies $frac {10}7$ of the distance the car was away. And each trip the car is $frac {3}{7}$ of the distance it was before. So the distance the fly travels is $sum_{k=0}^{infty} frac {10}7*(frac {3}{7})^k*20$ which if I did this correctly is



          $frac {10}7*20(sum_{k=0}^{infty} (frac {3}{7})^k)= frac {200}{7}frac 1{1-frac {3}{7}} = frac {200}{7}frac {7}{4}= 50$km.



          [1](for the record, in this time, $frac 1{70}h$ hours, the car has traveled $frac 47h$ and the fly has travelled $frac {10}7h$).







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 10 hours ago

























          answered 10 hours ago









          fleabloodfleablood

          73.4k22791




          73.4k22791























              15












              $begingroup$

              It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.



              The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                I got that but can you please give me more rigorous proof? If you don't mind.
                $endgroup$
                – user654700
                11 hours ago










              • $begingroup$
                @user654700 What exactly do you find not rigorous enough?
                $endgroup$
                – Arthur
                11 hours ago






              • 8




                $begingroup$
                @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
                $endgroup$
                – Ross Millikan
                11 hours ago












              • $begingroup$
                I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
                $endgroup$
                – inavda
                11 hours ago










              • $begingroup$
                Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
                $endgroup$
                – fleablood
                11 hours ago
















              15












              $begingroup$

              It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.



              The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                I got that but can you please give me more rigorous proof? If you don't mind.
                $endgroup$
                – user654700
                11 hours ago










              • $begingroup$
                @user654700 What exactly do you find not rigorous enough?
                $endgroup$
                – Arthur
                11 hours ago






              • 8




                $begingroup$
                @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
                $endgroup$
                – Ross Millikan
                11 hours ago












              • $begingroup$
                I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
                $endgroup$
                – inavda
                11 hours ago










              • $begingroup$
                Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
                $endgroup$
                – fleablood
                11 hours ago














              15












              15








              15





              $begingroup$

              It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.



              The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.






              share|cite|improve this answer









              $endgroup$



              It has made infinitely many trips. Every trip will be shorter than the last, but the fly will always reach the wall before the car, so it will always have room for one more trip. And one more. And one more.



              The total length the fly flies is 50km, as the car crashes into the wall exactly 30 minutes after the whole experiment started.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 11 hours ago









              ArthurArthur

              120k7119203




              120k7119203












              • $begingroup$
                I got that but can you please give me more rigorous proof? If you don't mind.
                $endgroup$
                – user654700
                11 hours ago










              • $begingroup$
                @user654700 What exactly do you find not rigorous enough?
                $endgroup$
                – Arthur
                11 hours ago






              • 8




                $begingroup$
                @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
                $endgroup$
                – Ross Millikan
                11 hours ago












              • $begingroup$
                I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
                $endgroup$
                – inavda
                11 hours ago










              • $begingroup$
                Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
                $endgroup$
                – fleablood
                11 hours ago


















              • $begingroup$
                I got that but can you please give me more rigorous proof? If you don't mind.
                $endgroup$
                – user654700
                11 hours ago










              • $begingroup$
                @user654700 What exactly do you find not rigorous enough?
                $endgroup$
                – Arthur
                11 hours ago






              • 8




                $begingroup$
                @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
                $endgroup$
                – Ross Millikan
                11 hours ago












              • $begingroup$
                I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
                $endgroup$
                – inavda
                11 hours ago










              • $begingroup$
                Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
                $endgroup$
                – fleablood
                11 hours ago
















              $begingroup$
              I got that but can you please give me more rigorous proof? If you don't mind.
              $endgroup$
              – user654700
              11 hours ago




              $begingroup$
              I got that but can you please give me more rigorous proof? If you don't mind.
              $endgroup$
              – user654700
              11 hours ago












              $begingroup$
              @user654700 What exactly do you find not rigorous enough?
              $endgroup$
              – Arthur
              11 hours ago




              $begingroup$
              @user654700 What exactly do you find not rigorous enough?
              $endgroup$
              – Arthur
              11 hours ago




              8




              8




              $begingroup$
              @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
              $endgroup$
              – Ross Millikan
              11 hours ago






              $begingroup$
              @user654700: This is rigorous. One can also write an equation for the length of each trip the fly makes and sum the series, but under the assumptions that is not needed. Don't confuse rigor with fancy mathematical symbols. There is a famous anecdote about John von Neumann and this puzzle.
              $endgroup$
              – Ross Millikan
              11 hours ago














              $begingroup$
              I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
              $endgroup$
              – inavda
              11 hours ago




              $begingroup$
              I suppose there are some minor details that could be added to give it some more rigor (or at least make the rigor more obvious): when the fly is at the position of the car, it has to travel the same distance to get to the wall, but its speed is greater, so it will always reach first. Once the fly is at the wall, we know that the car is not yet at the wall, due to the above reasoning. Since the fly has nonzero speed and the car does not have "infinite" speed, the fly and the car will next meet at some point between their current positions (not at the wall). Repeat.
              $endgroup$
              – inavda
              11 hours ago












              $begingroup$
              Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
              $endgroup$
              – fleablood
              11 hours ago




              $begingroup$
              Well, one can figure that if the fly is at the wall and the car is a non-zero distance away that the fly flies to the car and that takes some non-zero time (they have a combined speed which is finite so to cover a distance takes non-zero time). The fly will then fly to the wall. As the fly is faster it reaches the wall first. The car is now a non-zero distance away. Thus is the same situation as before so by induction there will be an infinite number of trips . That is formal and rigorous.
              $endgroup$
              – fleablood
              11 hours ago











              3












              $begingroup$

              Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.



              Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.






              share|cite|improve this answer










              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$













              • $begingroup$
                But how many trips is this?
                $endgroup$
                – Kathy
                6 hours ago










              • $begingroup$
                Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
                $endgroup$
                – Guest3711
                6 hours ago








              • 2




                $begingroup$
                I guess what I'm saying is you should edit your answer to include this.
                $endgroup$
                – Kathy
                6 hours ago
















              3












              $begingroup$

              Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.



              Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.






              share|cite|improve this answer










              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$













              • $begingroup$
                But how many trips is this?
                $endgroup$
                – Kathy
                6 hours ago










              • $begingroup$
                Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
                $endgroup$
                – Guest3711
                6 hours ago








              • 2




                $begingroup$
                I guess what I'm saying is you should edit your answer to include this.
                $endgroup$
                – Kathy
                6 hours ago














              3












              3








              3





              $begingroup$

              Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.



              Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.






              share|cite|improve this answer










              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              $endgroup$



              Simple way to calculate it: It takes the car 30 minutes to travel the 20km to the wall at 40km/hr. The fly, traveling at 100km/hr will travel 50km in those same 30 minutes.



              Edit: There will be an infinite number of trips. It's similar to how Zeno's paradox works where the trips get shorter and shorter and eventually take an infinitely small amount of time. But all those infinitely small trips end up as a finite amount of distance.







              share|cite|improve this answer










              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.









              share|cite|improve this answer



              share|cite|improve this answer








              edited 6 hours ago





















              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.









              answered 7 hours ago









              Guest3711Guest3711

              312




              312




              New contributor




              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.





              New contributor





              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.






              Guest3711 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.












              • $begingroup$
                But how many trips is this?
                $endgroup$
                – Kathy
                6 hours ago










              • $begingroup$
                Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
                $endgroup$
                – Guest3711
                6 hours ago








              • 2




                $begingroup$
                I guess what I'm saying is you should edit your answer to include this.
                $endgroup$
                – Kathy
                6 hours ago


















              • $begingroup$
                But how many trips is this?
                $endgroup$
                – Kathy
                6 hours ago










              • $begingroup$
                Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
                $endgroup$
                – Guest3711
                6 hours ago








              • 2




                $begingroup$
                I guess what I'm saying is you should edit your answer to include this.
                $endgroup$
                – Kathy
                6 hours ago
















              $begingroup$
              But how many trips is this?
              $endgroup$
              – Kathy
              6 hours ago




              $begingroup$
              But how many trips is this?
              $endgroup$
              – Kathy
              6 hours ago












              $begingroup$
              Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
              $endgroup$
              – Guest3711
              6 hours ago






              $begingroup$
              Infinite number of trips. Eventually the trips will be infinitely short and take an infinitely small amount of time. It's sort of like Zeno's paradox.
              $endgroup$
              – Guest3711
              6 hours ago






              2




              2




              $begingroup$
              I guess what I'm saying is you should edit your answer to include this.
              $endgroup$
              – Kathy
              6 hours ago




              $begingroup$
              I guess what I'm saying is you should edit your answer to include this.
              $endgroup$
              – Kathy
              6 hours ago











              1












              $begingroup$

              Arthur's answer is very good, but here is another (equally valid) way of visualizing the problem:



              Let us graph the positions of the wall, fly, and car over time. The wall doesn't move, so it is represented by a horizontal line. The car starts at some distance away from the wall but moves towards it at a constant speed until it hits the wall -- so we have a line that intersects the wall's line. Now for the interesting part:



              The fly's path starts off as a line with greater slope than the car's line, until it hits the wall's line. The fly's speed remains the same, but it is going in the opposite direction. So now the path continues as if the wall's line was a mirror and it was reflected. When the fly hits the car, the same thing happens -- the fly's path is reflected and it continues towards the wall again (one caveat: when it bounces off the car's line, the angles of incidence and reflection are not equal so it isn't behaving exactly as light would).



              So we can see that the fly's path continues bouncing up and down, always at the same or opposite slope.



              The final thing to notice to grasp the intuition is that this diagram we have constructed is self-similar. If we zoom in so that the second bounce with the wall is where the first bounce used to be, we have the same exact diagram as before. I won't prove this, but if you draw it out, you can see it intuitively. Essentially, this means that no matter how close the car gets to the wall, we can zoom in and see more bounces for the fly.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                Arthur's answer is very good, but here is another (equally valid) way of visualizing the problem:



                Let us graph the positions of the wall, fly, and car over time. The wall doesn't move, so it is represented by a horizontal line. The car starts at some distance away from the wall but moves towards it at a constant speed until it hits the wall -- so we have a line that intersects the wall's line. Now for the interesting part:



                The fly's path starts off as a line with greater slope than the car's line, until it hits the wall's line. The fly's speed remains the same, but it is going in the opposite direction. So now the path continues as if the wall's line was a mirror and it was reflected. When the fly hits the car, the same thing happens -- the fly's path is reflected and it continues towards the wall again (one caveat: when it bounces off the car's line, the angles of incidence and reflection are not equal so it isn't behaving exactly as light would).



                So we can see that the fly's path continues bouncing up and down, always at the same or opposite slope.



                The final thing to notice to grasp the intuition is that this diagram we have constructed is self-similar. If we zoom in so that the second bounce with the wall is where the first bounce used to be, we have the same exact diagram as before. I won't prove this, but if you draw it out, you can see it intuitively. Essentially, this means that no matter how close the car gets to the wall, we can zoom in and see more bounces for the fly.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Arthur's answer is very good, but here is another (equally valid) way of visualizing the problem:



                  Let us graph the positions of the wall, fly, and car over time. The wall doesn't move, so it is represented by a horizontal line. The car starts at some distance away from the wall but moves towards it at a constant speed until it hits the wall -- so we have a line that intersects the wall's line. Now for the interesting part:



                  The fly's path starts off as a line with greater slope than the car's line, until it hits the wall's line. The fly's speed remains the same, but it is going in the opposite direction. So now the path continues as if the wall's line was a mirror and it was reflected. When the fly hits the car, the same thing happens -- the fly's path is reflected and it continues towards the wall again (one caveat: when it bounces off the car's line, the angles of incidence and reflection are not equal so it isn't behaving exactly as light would).



                  So we can see that the fly's path continues bouncing up and down, always at the same or opposite slope.



                  The final thing to notice to grasp the intuition is that this diagram we have constructed is self-similar. If we zoom in so that the second bounce with the wall is where the first bounce used to be, we have the same exact diagram as before. I won't prove this, but if you draw it out, you can see it intuitively. Essentially, this means that no matter how close the car gets to the wall, we can zoom in and see more bounces for the fly.






                  share|cite|improve this answer









                  $endgroup$



                  Arthur's answer is very good, but here is another (equally valid) way of visualizing the problem:



                  Let us graph the positions of the wall, fly, and car over time. The wall doesn't move, so it is represented by a horizontal line. The car starts at some distance away from the wall but moves towards it at a constant speed until it hits the wall -- so we have a line that intersects the wall's line. Now for the interesting part:



                  The fly's path starts off as a line with greater slope than the car's line, until it hits the wall's line. The fly's speed remains the same, but it is going in the opposite direction. So now the path continues as if the wall's line was a mirror and it was reflected. When the fly hits the car, the same thing happens -- the fly's path is reflected and it continues towards the wall again (one caveat: when it bounces off the car's line, the angles of incidence and reflection are not equal so it isn't behaving exactly as light would).



                  So we can see that the fly's path continues bouncing up and down, always at the same or opposite slope.



                  The final thing to notice to grasp the intuition is that this diagram we have constructed is self-similar. If we zoom in so that the second bounce with the wall is where the first bounce used to be, we have the same exact diagram as before. I won't prove this, but if you draw it out, you can see it intuitively. Essentially, this means that no matter how close the car gets to the wall, we can zoom in and see more bounces for the fly.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 11 hours ago









                  inavdainavda

                  1177




                  1177






























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