Einstein in the boat, hat in the river: A grade-school problem
Ashok
Einstein was rowing a boat in a river upstream when the
boat passed under a bridge, knocking off his hat, which
drifted downstream. Being absent-minded he noticed it
only after 5 minutes had elapsed. He quickly turned his
boat around and rowed with equal effort in pursuit of
the hat. He caught it 1 km from the bridge. What was
the speed of the river?
1. It's easy enough to solve with a bit of algebraic
notation, but I am looking to see if there is an
intuitive verbal argument that a grade-school student
would find perspicuous.
2. Inititally, I thought it should be very easy to
prove geometrically: Plot time on the x-axis and
distance from the bridge on the y-axis, say, with
positive numbers standing for movement in the
direction of the river flow. Draw a line from the
origin in the first quadrant (NE) for the movement
of the hat (line 1). Draw a line from the origin
in the fourth quadrant (SE) for the movement of
the boat (line 2a). Then draw the movement of the
boat downhill by changing the orientation by an
appropriate angle (line 2b). Then try to prove
that the point where lines 1 and 2b intersect
is twice the distance from the point where the
boat turns. I couldn't do it readily. Can it be
done using only school geometry?
Ashok
Jack D
> Here's a question that showed up in my niece's homework:
>
> Einstein was rowing a boat in a river upstream when the
> boat passed under a bridge, knocking off his hat, which
> drifted downstream. Being absent-minded he noticed it
> only after 5 minutes had elapsed. He quickly turned his
> boat around and rowed with equal effort in pursuit of
> the hat. He caught it 1 km from the bridge. What was
> the speed of the river?
As the hat floats it can be considered to move at the same speed of the
river. The puzzle states that Einstein rowed with equal effort, meaning his
speed was equal relative to the water, so he must have caught the hat after
another 5 minutes have elapsed.
This means the hat has moved 1km in 10 minutes, giving the hat (and the
river) a speed of 6 km/h.
If you get the reasoning correct the algebra is simple.
Art Neuendorffer
>>Here's a question that showed up in my niece's homework:
>>
>>Einstein was rowing a boat in a river upstream when the
>>boat passed under a bridge, knocking off his hat, which
>>drifted downstream. Being absent-minded he noticed it
>>only after 5 minutes had elapsed. He quickly turned his
>>boat around and rowed with equal effort in pursuit of
>>the hat. He caught it 1 km from the bridge. What was
>>the speed of the river?
Jack D wrote:
> As the hat floats it can be considered to move at the same speed of the
> river. The puzzle states that Einstein rowed with equal effort, meaning his
> speed was equal relative to the water, so he must have caught the hat after
> another 5 minutes have elapsed.
>
> This means the hat has moved 1km in 10 minutes, giving the hat (and the
> river) a speed of 6 km/h.
Is the 5/10 minutes measured by Einstein's own watch?
The Twin Paradox says that the hat could have been floating for days!
Art Neuendorffer
Jack D
Yes it does! But I am putting my money on the fact that he is looking at his
own watch. And even if it was a church tower or something with a clock that
he saw in the distance, I assume that the speed at which he was rowing (!)
wouldn't have been of such magnitude that the influence of the Twin paradox
would be big enough. It might differ a fraction of the speed though, but
then again the ripples caused by the falling had might have a reverse
effect.
Art Neuendorffer
>>
>>>>Here's a question that showed up in my niece's homework:
>>>>
>>>>Einstein was rowing a boat in a river upstream when the
>>>>boat passed under a bridge, knocking off his hat, which
>>>>drifted downstream. Being absent-minded he noticed it
>>>>only after 5 minutes had elapsed. He quickly turned his
>>>>boat around and rowed with equal effort in pursuit of
>>>>the hat. He caught it 1 km from the bridge. What was
>>>>the speed of the river?
>>>
>>Jack D wrote:
>>
>>
>>>As the hat floats it can be considered to move at the same speed of the
>>>river. The puzzle states that Einstein rowed with equal effort, meaning his
>>>speed was equal relative to the water, so he must have caught the hat after
>>>another 5 minutes have elapsed.
>>>
>>>This means the hat has moved 1km in 10 minutes, giving the hat (and the
>>>river) a speed of 6 km/h.
> Art Neuendorffer wrote on 26-11-2002 14:41:
>> Is the 5/10 minutes measured by Einstein's own watch?
>>
>> The Twin Paradox says that the hat could have been floating for days!
Jack D wrote:
> Yes it does! But I am putting my money on the fact that
> he is looking at his own watch.
That's the problem: if he is, indeed, the time keeper looking
at his own watch than a Twin Paradox correction must be applied.
> And even if it was a church tower or something with a clock that
> he saw in the distance, I assume that the speed at which he was rowing (!)
> wouldn't have been of such magnitude that the influence of the Twin paradox
> would be big enough.
If Einstein, the time keeper, is watching a clock tower behind him
(he is rowing after all) then the situation is *even worse* than
the Twin paradox would give. At barely half the speed of light
the 5 minute clock signal wouldn't reach Einstein
(5 light minutes away) until 10 minutes had passed
(bridge/hat time).
> It might differ a fraction of the speed though, but
> then again the ripples caused by the falling might
> have a reverse effect.
It would certainly help if this all took place in a world Einstein
probably imagined where even Zurich's trolley approached light speed.
Art Neuendorffer
Eric Nielsen
> boat passed under a bridge, knocking off his hat, which
> drifted downstream. Being absent-minded he noticed it
> only after 5 minutes had elapsed. He quickly turned his
> boat around and rowed with equal effort in pursuit of
> the hat. He caught it 1 km from the bridge. What was
> the speed of the river?
> 1. It's easy enough to solve with a bit of algebraic
> notation, but I am looking to see if there is an
> intuitive verbal argument that a grade-school student
> would find perspicuous.
There is another guy floating on the river in an inner tube relaxing. For
all he knows, he's sitting still and bridge and the shore are moving past
him. To him the water is not moving. He sees Einstein row towards him, and
they pass under the bridge. Now he sees the bridge and Einstein moving
away, but Einstein's hat is sitting in the water next to him. He watches
Einstein paddle away from him for 5 minutes, and then paddle towards him for
5 minutes. When Einstein reaches him, he looks up and the bridge is 1 km
away. He figures that if the bridge can move 1 km in 10 minutes, it can
move 6 km in 60 minutes.
Cheers,
Eric
Petter Solbu
> As the hat floats it can be considered to move at the same speed of the
> river. The puzzle states that Einstein rowed with equal effort, meaning
his
> speed was equal relative to the water, so he must have caught the hat
after
> another 5 minutes have elapsed.
I don't believe this is correct. Notice that he finds the hat 1km FROM the
bridge downstream. It is therefore no reason to believe that only 5 minutes
have elapsed. 5 minutes have elapsed when Einstein cross the bridge again,
but not when he finds the hat.
PS
Petter Solbu
> There is another guy floating on the river in an inner tube relaxing. For
> all he knows, he's sitting still and bridge and the shore are moving past
> him. To him the water is not moving. He sees Einstein row towards him,
and
> they pass under the bridge. Now he sees the bridge and Einstein moving
> away, but Einstein's hat is sitting in the water next to him. He watches
> Einstein paddle away from him for 5 minutes, and then paddle towards him
for
> 5 minutes.
After Einstein's return it takes 5 minutes for Einstein to reach the bridge,
not the hat. We do not know how fast Einstein is rowing comparised with the
river speed.
> When Einstein reaches him, he looks up and the bridge is 1 km
> away. He figures that if the bridge can move 1 km in 10 minutes, it can
> move 6 km in 60 minutes.
I do not believe this is correct.
PS
Art Neuendorffer
>
>>As the hat floats it can be considered to move at the same speed of the
>>river. The puzzle states that Einstein rowed with equal effort, meaning
>> his speed was equal relative to the water, so he must have caught
>> the hat after another 5 minutes have elapsed.
Petter Solbu wrote:
> I don't believe this is correct. Notice that he finds the hat 1km FROM the
> bridge downstream. It is therefore no reason to believe that only 5 minutes
> have elapsed. 5 minutes have elapsed when Einstein cross the bridge again,
> but not when he finds the hat.
If Einstein is rowing back *with* the stream then he passes the
bridge in less than 5 minutes.
If Einstein is rowing back *against* the stream then he hasn't yet
reached the bridge in the allotted 5 minutes.
However, in the Galilean reference frame in which both hat & stream
are fixed it takes Einstein the same time to return to the hat as it
took for him to row away: 5 minutes.
Art
Petter Solbu
> However, in the Galilean reference frame in which both hat & stream
> are fixed it takes Einstein the same time to return to the hat as it
> took for him to row away: 5 minutes.
Why? Is this based on a physical formula or something?
PS
Art Neuendorffer
>> However, in the Galilean reference frame in which both hat & stream
>>are fixed it takes Einstein the same time to return to the hat as it
>>took for him to row away: 5 minutes.
Petter Solbu wrote:
> Why? Is this based on a physical formula or something?
It is based on physical logic:
1) the theory of Relativity says that any unaccelerated
frame of reference is equally valid.
2) a rower can go in any direction
with equal speed in still water
(ignoring wind resistance).
Art
Petter Solbu
> 1) the theory of Relativity says that any unaccelerated
> frame of reference is equally valid.
Ok. I am not into physics, so you have to excuse me. :-) But I just wonder:
The puzzle explained that Einstein returned and rowed in equal effort. I
thought this meant the same speed. What does it mean really?
> 2) a rower can go in any direction
> with equal speed in still water
> (ignoring wind resistance).
But this is not still water. How does this fit into the context then?
PS
Art Neuendorffer
>>1) the theory of Relativity says that any unaccelerated
>> frame of reference is equally valid.
Petter Solbu wrote:
> Ok. I am not into physics, so you have to excuse me. :-) But I just wonder:
> The puzzle explained that Einstein returned and rowed in equal effort. I
> thought this meant the same speed. What does it mean really?
It means the same speed with respect to the current.
>>2) a rower can go in any direction
>> with equal speed in still water
>> (ignoring wind resistance).
> But this is not still water. How does this fit into the context then?
For simplicity, a moving frame of reference was chosen in which the hat
& the stream are not moving (but the bridge is). In this frame of
reference Einstein can row away for 5 minutes (thru still water) turn
around and row back in 5 minutes( thru still water). When he returns to
the fixed hat after 10 minutes *the bridge* has drifted 1 km. away.
You can chose more complicated frames of reference (e.g., the bridge
is fixed, the moon is fixed, Alpha Centauri is fixed. . .) and still
solve the problem but the frame in which the hat & stream are not moving
is the only one simple enough to solve the problem in one's head. It's
just that now it is the bridge that has drifted 1 km.
Art
Petter Solbu
> > Ok. I am not into physics, so you have to excuse me. :-) But I just
wonder:
> > The puzzle explained that Einstein returned and rowed in equal effort. I
> > thought this meant the same speed. What does it mean really?
>
> It means the same speed with respect to the current.
So if the speed was 10 km/h upstream, it is 10 km/h downstream too? If that
is correct, I do not understand why he uses the same time away from the hat
than he uses towards it.
> For simplicity, a moving frame of reference was chosen in which the hat
> & the stream are not moving (but the bridge is). In this frame of
> reference Einstein can row away for 5 minutes (thru still water) turn
> around and row back in 5 minutes( thru still water). When he returns to
> the fixed hat after 10 minutes *the bridge* has drifted 1 km. away.
But I can't understand why he reach the hat after another 5 minutes if the
speed is the same upstream and downstream.
PS
Art Neuendorffer
>>>The puzzle explained that Einstein returned and rowed in equal effort.
>>> I thought this meant the same speed. What does it mean really?
> "Art Neuendorffer" wrote:
>
>> It means the same speed with respect to the current.
Petter Solbu wrote:
> So if the speed was 10 km/h upstream, it is 10 km/h downstream too? If that
> is correct,
Yes, with respect to the current or the hat;
but the speeds are different with respect to the bridge.
> I do not understand why he uses the same time away from the hat
> than he uses towards it.
If it were the bridge then it would indeed take longer one way than
the other. . .which is why he ends up 1 km. away from the bridge at the
end. However, the stream drift on the rowboat & the hat are identical
and therefore does not affect the *relative motion* between boat & hat.
>>For simplicity, a moving frame of reference was chosen in which the hat
>>& the stream are not moving (but the bridge is). In this frame of
>>reference Einstein can row away for 5 minutes (thru still water) turn
>>around and row back in 5 minutes( thru still water). When he returns to
>>the fixed hat after 10 minutes *the bridge* has drifted 1 km. away.
Petter Solbu wrote:
> But I can't understand why he reach the hat after another 5 minutes
> if the speed is the same upstream and downstream.
There *is no* upstream or downstream in the chosen frame of reference;
there is only *bridge drift* with respect to that frame of reference.
Astronauts in the Space Station do not generally care which direction
the earth is or how fast the earth is moving next to them; as far as
they are concerned the Space Station is standing still and the earth is
speeding past. An astronaut with a jet pack can take short trips in any
direction and return in the same amount of time (though the earth will
have drifted in the interim).
Art
Eric Nielsen
> Ok. I am not into physics, so you have to excuse me. :-) But I just
wonder:
> The puzzle explained that Einstein returned and rowed in equal effort. I
> thought this meant the same speed. What does it mean really?
Many years later, Einstein is much older and not into rowing, so he takes
the train everywhere. Einstein is walking through the train when it is
passing Tucamcari Station. His hat falls off and lands on the floor of the
train car, but he doesn't notice. He keeps walking through the train for
five minutes (it's a very long train), when he notices his lack of hat. He
turns around and walks back through the train *with equal effort* until he
gets back to where his hat fell off. He askes the conductor where they are,
and the conductor tells him they are 1km past Tucamcari Station.
Just as "walking with equal effort" to the front or back of a train means
walking with the same speed relative to the train, "rowing with equal
effort" upstream or downstream means rowing with equal speed relative to the
water (or relative to something floating in the water)
Hope this helps,
Eric
Art Neuendorffer
>
> wonder:
>
>>The puzzle explained that Einstein returned and rowed in equal effort.
>> I thought this meant the same speed. What does it mean really?
>
> Many years later, Einstein is much older and not into rowing, so he takes
> the train everywhere. Einstein is walking through the train when it is
> passing Tucamcari Station. His hat falls off and lands on the floor of the
> train car, but he doesn't notice. He keeps walking through the train for
> five minutes (it's a very long train), when he notices his lack of hat. He
> turns around and walks back through the train *with equal effort* until he
> gets back to where his hat fell off. He askes the conductor where they are,
> and the conductor tells him they are 1km past Tucamcari Station.
Unfortunately, distances to & from Tucumcari are *always*
measured in "miles" in order to keep the meter:
Twelve more miles to Tucumcari
I've been hurrying there
To the gal I'm gonna marry
With the yellowest hair
{Left right march along
I've just gotta get home} (just gotta get home)
Ten more miles to Tucumcari
Then I'll never more roam
(Tucumcari, Tucumcari, I just gotta get home)
Eight more miles to Tucumcari
It's the fourth of July
Been three years in January
Since I kissed her goodbye
{Left right march along
I've just gotta get home} (just gotta get home)
Six more miles to Tucumcari
Then I'll never more roam
(Tucumcari, Tucumcari, I just gotta get home)
Four more miles to Tucumcari
Not much further to go
Got no time to waste or tarry
She'll be waitin' I know
{Left right march along
I've just gotta get home} (just gotta get home)
Two more miles to Tucumcari
Then I'll never more roam
(Tucumcari, Tucumcari, I just gotta get home)
Here I am in Tucumcari
Found my yellow-haired gal
Just in time to see her marry
With my very best pal
{Left right march along
Guess I better be gone} (I better be gone)
A thousand miles from Tucumcari
I'll be rambling on
(Tucumcari, Tucumcari, I'll be rambling on)
A hundred miles from Tucumcari
Down in Santa Fe
Fell in love with a girl named Mary
Gee I'm happy today
{No more marching now
Found my honeycomb} (found my honeycomb)
I've forgotten Tucumcari
No more reason to roam
(Tucumcari, Tucumcari)
I've forgotten Tucumcari
No more reason to roooaaaammmmm
Eric Nielsen wrote:
> Just as "walking with equal effort" to the front or back of a train means
> walking with the same speed relative to the train, "rowing with equal
> effort" upstream or downstream means rowing with equal speed relative
> to the water (or relative to something floating in the water)
Also it was not Einstein's hat but rather
Shelley Winters who fell into the water.
(Ramond Burr was the first to solve the puzzle.)
Art Neuendorffer
Petter Solbu
> Many years later, Einstein is much older and not into rowing, so he takes
> the train everywhere. Einstein is walking through the train when it is
> passing Tucamcari Station. His hat falls off and lands on the floor of
the
> train car, but he doesn't notice. He keeps walking through the train for
> five minutes (it's a very long train), when he notices his lack of hat.
He
> turns around and walks back through the train *with equal effort* until he
> gets back to where his hat fell off. He askes the conductor where they
are,
> and the conductor tells him they are 1km past Tucamcari Station.
>
> Just as "walking with equal effort" to the front or back of a train means
> walking with the same speed relative to the train, "rowing with equal
> effort" upstream or downstream means rowing with equal speed relative to
the
> water (or relative to something floating in the water)
Of course! Now I understand. Thank you very much.
PS
Eric Nielsen
> measured in "miles" in order to keep the meter:
A brief statistics/history page about Tucumcari:
http://www.pe.net/~rksnow/nmcountytucumcari.htm
Interesting...the area of Tucumcari is only given in square kilometers, but
the distances from Tucumcari to Santa Fe and DC are only given in miles.
Something much more sinister than simple artistic license may be afoot here.
Any New Mexicans care to elucidate?
Surendar Jeyadev
Wait till he meets his twin rowing the other way ...
--
Surendar Jeyadev jey...@wrc.xerox.bounceback.com
Remove 'bounceback' for email address