[Guest]

This is a fairly easy puzzle but I'm looking for the right thought process

I was rowing a boat upstream at a speed of 3 miles/hr (the best I could do) on a river flowing at 5 miles/hr. I dropped my hat off and realised it after 35 minutes. So, I immediately turned around and rowed my boat with the 'same effort' that I had put in to go up the stream. How long will it take me to reach my hat, if I ever will. (Please note that the hat is constantly moving away with the river)

[Guest]

35 minutes ... the river's speed is constant

edit(my thinking process was wrong)

Edited June 5, 2010 by PVRoot

[Glycereine]

If you mean you are going 3 mph compared to shore, then you are going 8 mph waterspeed. If this is the case, then after 35 minutes your hat is 8 mph*35min/60 min/hr) away from you and you will have to travel the same time at the same speed relative to the river to catch back up. so 35 minutes.

However if you are traveling at 3mph compared to the river, you will never catch your hat.

Edited June 5, 2010 by Glycereine

[Guest]

This is a fairly easy puzzle but I'm looking for the right thought process

I was rowing a boat upstream at a speed of 3 miles/hr (the best I could do) on a river flowing at 5 miles/hr. I dropped my hat off and realised it after 35 minutes. So, I immediately turned around and rowed my boat with the 'same effort' that I had put in to go up the stream. How long will it take me to reach my hat, if I ever will. (Please note that the hat is constantly moving away with the river)

Probably many answers from others already, but I won't look yet.

Ignoring relativity, and assuming the stream is infinitely long and uniform,

At each moment, from the reference point of the hat, you travel 8 mph away from it before you turn around.

If you can travel 3mph against 5mph of flow, then relative to still water you can travel 8mph. If the water is flowing with you at 5mph, then you can travel 13 mph relative to still water. From the point of view of the hat, you are now travelling 8 mph towards it. Thus to undo your previous action from the frame of reference of the hat: "rowing 8mph away for 35 min", you must perform the action "row 8mph toward for X min". Clearly X is 35.

Thus it should take the guy 35 min to get back to his hat.

[Guest]

Oops I did not put it in spoiler ... read the next post

Edited June 5, 2010 by Me&MyBrain

[Guest]

Well, In 35 minutes the distance between you and your hat would be 8 mph (your speed of 3 mph + river's speed of 5 mph) * 35 minutes = 8/60 * 35 = 14 / 3 miles. Now you have to travel 14/3 miles to get to your hat with a speed of 3 mph (the 5 mph speed river provides is the same for you and the hat and so that cancels out) so it would take 93.333 minutes to get to the hat.

[Guest]

If you mean you are going 3 mph compared to shore, then you are going 8 mph waterspeed. If this is the case, then after 35 minutes your hat is 8 mph*35min/60 min/hr) away from you and you will have to travel the same time at the same speed relative to the river to catch back up. so 35 minutes.

However if you are traveling at 3mph compared to the river, you will never catch your hat.

If you are going 3 mph compared to the river, then you are travelling backwards at 2mph with respect to the shore (moving in the same direction as your hat, but facing the wrong way. Your efforts are overpowered by the river, but your capability is to move 3mph with respect to still water. If you turn around and go with the river, you will now travel at 8mph w/ respect to the shore towards your hat.

With respect to the river, you are travelling 3mph in the direction of flow.

With respect to the river/hat, you initially traveled in one direction for 35 min at a speed of 3mph. Then you turn around and travel at a speed of 3mph in the other direction.

The hat represents your starting point.

To get back to the starting point, you must once again travel for 35 minutes.

In either case, I think it is 35 minutes.

The only weakness to the argument that I know of is that strictly speaking, you cannot simply add velocities from different frames of reference. This is called the Galilean transformation, and is true in a practical sense for every day objects.

Einstein showed that there is actually an additional factor that is undetectable for every day objects and actions, but is significantly detectable when your objects are moving extremely fast (unlike our rowing boats). Look up Lorentz transformations if you are interested. Applying the Lorentz transformation to the problem is silly though, as the difference in calculation would be negligible, and there are many more obvious and significant improvements to the rowboat model described above that should be implemented before considering relativity.

[Guest]

While returning your effor would produce 13 mph of which 5 mph should be the same for you and the hat. So You will be approaching the hat at 8 mph which is the same speed with which you were moving away from the hat. So, it should take you the same time of 35 minutes to reach the hat.

Edited June 5, 2010 by Me&MyBrain

[Guest]

Well, In 35 minutes the distance between you and your hat would be 8 mph (your speed of 3 mph + river's speed of 5 mph) * 35 minutes = 8/60 * 35 = 14 / 3 miles. Now you have to travel 14/3 miles to get to your hat with a speed of 3 mph (the 5 mph speed river provides is the same for you and the hat and so that cancels out) so it would take 93.333 minutes to get to the hat.

When you turn around to go with the river, you will go much faster than 3mph.

You will go 13 mph because the river works with you, not against you.

The distance between you and your hat will then have a rate

5mph (from hat) - 13 mph (your speed) = -8mph

The time for the distance to reach zero is therefore

(14/3 mi)/(8mph) = 7/12 hours = 35 min

Edit: You got the correction before me.

Edited June 5, 2010 by mmiguel1

[Guest]

When you turn around to go with the river, you will go much faster than 3mph.

You will go 13 mph because the river works with you, not against you.

The distance between you and your hat will then have a rate

5mph (from hat) - 13 mph (your speed) = -8mph

The time for the distance to reach zero is therefore

(14/3 mi)/(8mph) = 7/12 hours = 35 min

Edit: You got the correction before me.

Thanks.

[Guest]

At the time you realize you dropped your hat, you and the hat are 49/12 miles apart. (Reason: you travel at 2 mph upstream and the hat travels at 5 mph with the stream for the 35 minutes)

The time starts as soon as you turn around. When you turn around, you now travel at 8 mph and the hat still at 5 mph. From turn around point, your total distance until you reach the hat is 8t. That distance is equal to the distance the hat travelled (dropping point to finding point) plus the 49/12 miles it was from you. So 8t = 5t + 49/12… t= 49/36 hours… which is 1 hour 21 minutes and 40 seconds.

[Guest]

I meant to say "

The time starts as soon as you turn around. When you turn around, you now travel at 8 mph and the hat still at 5 mph. From turn around point, your total distance until you reach the hat is 8t. That distance is equal to the distance the hat travelled (starting time to finding point) plus the 49/12 miles it was from you. So 8t = 5t + 49/12… t= 49/36 hours… which is 1 hour 21 minutes and 40 seconds."

Edited June 6, 2010 by MCM1984

[Guest]

A lot of people got the correct answer and it indeed is the same time i.e. 35 minutes

Here's just a simpler way to look at the problem: The speeds mentioned above - speed of the river and mine were obviously relative to the land. But, If you take the river as your frame of reference, then the speeds become immaterial. Also, upstream/downstream does not matter because I used the same effort in both directions.

So, I am on a river and row in one direction - drop my hat - keep rowing for x hrs - then turn around and row with the same effort - It will take me the same amount of time to reach my hat because obviously the speed of hat relative to the water is 0

[Guest]

9 minutes.

[Guest]

There are two other considerations I'd like to add to this discussion:

Firstly, the hat will not go as fast as the current, due to the drag effect.

Secondly, the hat might sink, and you would go past it without seeing it!

[Guest]

i think you're wrong HB, here's why.

although your answer seems logical, here's the problem.

because of the flow of the river, your going a different velocity one direction compared to the other.

when traveling up river, you're going 8 miles an hour away from the hat.

(8(miles per hour) *35(minutes)/60(minutes per hour) = 4.666(miles))

when going toward the hat you're going 3 miles an hour. (both you and hat traveling 5 miles an hour.)

(4.666(miles)/3(miles per hour) = 1.555(hours) = 93.333(minutes))

[Guest]

let's put this problem a different way. imagine Car A travels at 5.5 miles an hour, and car B travels at 2.5 miles an hour. they start off traveling away from each other, for 35 minutes. then car A turns around and heads toward car B.

even though car B's speed is constant, there will be a difference in the amount of time going away from and coming to.

[Guest]

let's put this problem a different way. imagine Car A travels at 5.5 miles an hour, and car B travels at 2.5 miles an hour. they start off traveling away from each other, for 35 minutes. then car A turns around and heads toward car B.

even though car B's speed is constant, there will be a difference in the amount of time going away from and coming to.

The car analogy would only apply if the river acted like a road (if the river was not flowing).

I think a better analogy might be one of those moving walkways (I've seen them at airports).

When you are walking against the walkway, you are subtracting the speed of the walkway from your own walking speed (you go much slower). When you are walking with the walkway, you add the speed of the walkway to your own (you go much faster).

If your walking speed is W, and the speed that the ground (the walkway) is moving is G, then going against the walkway gives a speed of W-G, and going with the walkway gives a speed of W+G.

The hat is always moving in the same direction with a speed of G.

When walking away from your hat, the rate change in the distance between you and your hat is (W-G) + G = W.

W-G is your net speed away from point of dropping hat(your walking speed - penalty factor of the ground underneath you moving the opposite direction).

+G is the speed of the hat moving away (and increasing distance).

When walking toward your hat, the rate of change is -(W+G)+G = -W

-(W+G) is net in direction towards hat (+G because ground is moving in same direction as you now).

+G is still the speed of the hat moving away.

The rates are the same magnitude going either direction, and require the same time.

The key idea here is that your ability to maintain speed depends on the motion of your immediate surroundings.

If you are in a slow moving train, and you are walking from the front of the train to the back, you will to an outside observer, appear to be moving slower than a person of similar walking speed walking on the ground next to the train.

On the other hand, if you were walking from the back of the train to the front, you would appear to walk much faster than a person on the ground.

To apply this to your previous reasoning,

when you are moving upriver, the distance between you and the hat is increasing by 8mph. True. W=8.

To ensure clarity, your actual boat speed is not 8 mph, but actually W-G = 3mph.

When you are moving downriver, the distance between you and the hat is 3mph. False. This assumes that when you turn around, your actual boat speed is 8mph. Your actual boat speed is W+G = 13 mph. This is a valid conclusion, because W (your boat's max speed) should not depend on the direction you are facing. You are now moving with the water, and just as if you were walking the same direction as a train, inside the train, you should now move faster.

The rate of decrease of the distance between you and the hat is 13-5 = 8 mph. Same as before.

[Guest]

A lot of people got the correct answer and it indeed is the same time i.e. 35 minutes

Here's just a simpler way to look at the problem: The speeds mentioned above - speed of the river and mine were obviously relative to the land. But, If you take the river as your frame of reference, then the speeds become immaterial. Also, upstream/downstream does not matter because I used the same effort in both directions.

So, I am on a river and row in one direction - drop my hat - keep rowing for x hrs - then turn around and row with the same effort - It will take me the same amount of time to reach my hat because obviously the speed of hat relative to the water is 0

For those who prefer numbers

Although, the above solution is the simplest way to look at the problem, here's an explaination for those mathematically inclined:

While going upstream, my speed relative to the Hat : 3 - (-5) = 8 miles/hr

Now please note that I am going upstream at 3 miles/hr. If the water were stationary and I used the same effort, I'd go at 8 miles/hr. Another way to say this is my speed relative to the water is 3 + 5 = 8miles/hr. Now if I use the same effort to row downstream,

my speed will be again 8 miles/hr relative to the water.

Therefore, my actual speed downstream is (i.e. relative to the land, our actual frame of reference here) = 8 + 5 = 13 miles/hr

My speed relative to the hat while going downstream = 13 - 5 = 8 miles/hr

Since my speed relative to the hat is the same - 8 miles/hr, in both directions, I will take the same time to reach my hat.

I hope this helps. If not, please see mmiguel1's explaination (Thanks!) which is more elaboarate.

[Guest]

There are two other considerations I'd like to add to this discussion:

Firstly, the hat will not go as fast as the current, due to the drag effect.

Secondly, the hat might sink, and you would go past it without seeing it!

@Donjar:

The puzzle assumes perfect conditions like most other puzzles.

There is no drag effect and the hat won't sink.

The speed of hat relative to the water is 0

There are no holes in the Hat

It will not drift to the bank.

Also, it assumes that the speed of the river is same in all planes, irrespective of the distance from the bank (which is again unreal)

The friction between all surfaces is equal

The time gap between Hat falling off and starting to flow with the river is 0

The gravitational pull is equivalent to that of the Earth

It also assumes, that wind speed is 0miles/hr and it's not raining/snowing etc etc.

Basically, it acts as a point object and so does my boat

I apologize for assuming.

[Guest]

@Donjar:

The puzzle assumes perfect conditions like most other puzzles.

There is no drag effect and the hat won't sink.

The speed of hat relative to the water is 0

There are no holes in the Hat

It will not drift to the bank.

Also, it assumes that the speed of the river is same in all planes, irrespective of the distance from the bank (which is again unreal)

The friction between all surfaces is equal

The time gap between Hat falling off and starting to flow with the river is 0

The gravitational pull is equivalent to that of the Earth

It also assumes, that wind speed is 0miles/hr and it's not raining/snowing etc etc.

Basically, it acts as a point object and so does my boat

I apologize for assuming.

For talking about assumptions like these, I always find it amusing to point out that we have left out the effect of the gravity of Pluto. There are so many simplifying assumptions we make in every aspect of thought, and sometimes it takes something as ridiculous as a challenge to factor in the gravity of Pluto to a silly toy problem to find the humility to accept that assumptions are necessary and important for drawing conclusions. So please, don't apologize.

Edited June 7, 2010 by mmiguel1