# Thread: The Boat and the Hat

1. ## Re: The Boat and the Hat

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>200 units</font color=yellow></span hi>

I did assume that the SPEED of the ROWING was constant and the VELOCITY did NOT remain constant but REVERSED when the boat was turned around.

A velocity is speed and DIRECTION. If you turn around the velocity MUST change since the direction changes!

Steve

2. ## The Boat and the Hat

A man is rowing his boat down stream. At some point, his hat gets knocked off into the water but he keeps rowing. Ten minutes later, he notices that his hat has been knocked into the water, turns around and rows up stream to retrieve it.

The river is flowing 10 units (inches, feet, meters, kilometers take your pick) per minute down stream. Assuming constant speed (edited to remove velocity) and the man has the amazing ability to turn his boat round in 0 seconds, HOW FAR HAS THE HAT TRAVELLED IN THE WATER?

3. ## Re: The Boat and the Hat

GRRR! One little word that I knew was technically incorrect when I wrote it. It seems I should have made the additional effort to put in SPEED.

Oh - and by the way, you are correct. Care to share how you worked it out?

4. ## Re: The Boat and the Hat

Define: R = Rowing Speed of boat (units/min)
x = time (minutes) from turning around until intercepts hat

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>The hat travels a total of 10+ x minutes so the distance from Start is (10+x) * 10
The boat travels from start to the point (note that the second part is NEGATIVE distance, boat is going back to start):
10 * (10 + B ) + x (10 - [img]/forums/images/smilies/cool.gif[/img]

When they intercept the distance (from start) will be the same!:
(10+x) * 10 = 10 (10 + B ) + x (10 - [img]/forums/images/smilies/cool.gif[/img]
100 + 10x = 100 + 10 B + 10x - Bx
0 = 10B - Bx
10B = Bx
10 = x

Hat travels 10 + x = 10 + 10 = 20 minutes, so travels 20 min * 10 units/min = 200 units</font color=yellow></span hi>

Steve

5. ## Re: The Boat and the Hat

The person who told me about this said that there wasn't an algebraic solution (for some unknown reason). I didn't have a pen and paper available, so I assumed <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow> that the rower could row at an incredible speed (10x speed of light). With that assumption, the speed of the rower is effectively not affected by the speed of the river, so the rows for 10 minutes down stream and 10 minutes back up stream - a total of 20 minutes. The hat travels 20 x 10 units in that time.</font color=yellow></span hi>

6. ## Re: The Boat and the Hat

What is interesting <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>is that NO MATTER what the speed of the river, and NO MATTER how long you paddle AWAY from the hat, It will take you EXACTLY how long you paddled AWAY to paddle back to meet your hat!

If you do all the algebra, you get that the time FORWARD without hat = time RETURNING without hat!
</font color=yellow></span hi>

7. ## Re: The Boat and the Hat

<span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>It's all a matter of relativity. The river is moving at a constant speed, so only consider the movement of the rower relative to the river. Without knowing the speed of the rower or the river, we can say thet the rower moves away from the hat for 10 minutes at a certain speed, then moves towards the hat at the same speed but reversed in direction, so it takes the same time. Perhaps imagining walking on a moving walkway or a conveyor belt make it easier to understand.</font color=yellow></span hi>

8. ## Re: The Boat and the Hat

Surely, if he rows at 10x speed of light, he'll end up passing himself as a small child, as he loses his hat in the first instance... or some equally paradoxical scenario. <img src=/S/grin.gif border=0 alt=grin width=15 height=15> Still, that's an interesting approach to solving it - somewhat akin to the hole in the sphere problem.

Alan

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