A gentleman and his yacht
A gentleman with a top hat is taking a ride in his fabulous yacht in a long river. In still water, the yacht moves at a constant velocity v . This can be affected by the direction and speed of the current. He sailed his yacht against the current of the river for 1 mile, at which point his hat blew off and landed in the river.
He didn't notice this for 30 minutes until it began to drizzle and he realized his hat must have blown off. Upon realizing this, he turned around, land embarked on a quest to retrieve his hat.
Amazingly, he reached the dock that he started from, at the same time his hat did! What was the rate of the current in miles per hour? (And yes, you have enough information)
This problem is not original.
The answer is 1.
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5 solutions
Very impressed by the way of solution. I would put it in another words. Since the man and the hat both are sailing in the river after the hat is blown off, the speed of the river can be any thing, but distance between them depends on the velocity of the man and his direction only. Again man's velocity remains constant. Since he is moving away for 30 minutes, he must travel for 30 minutes back to be with his hat again. Thus they were separated for 1 hour. But the hat had traveled for this one hour by 1 mile back with the velocity of the river. So river velocity is 1mile/1hour. The velocity of the man can be any thing!!! This is just same as we are on a luxury liner. The liner is moving with speed V. A man runs from the back to the front for 5 minutes to go to the starting place with the same speed, he will require to 5 minutes what ever be the speed V. By the time, if the liner has covered V miles, the liner speed is V miles per 10 minutes=6Vmles per hour..
The problem with this question is that if the boat and current are both at 1 mph, sailing against it, he travels at 0 mph, or does not move during his upriver portion. Also, the null solution works, ie he and the current are at 0 mph
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Notice that he must travel 1 mile, which does not happen in both of your cases as his net velocity is 0 in them.
Sam, the first part of the question said that the yacht sailed 1 mile. It did not, however, specify the time it took to sail that mile. Your assumption of 1 hour is not correct.
- The man travels one mile
- The man also travel 0.5*(Vy-Vc) miles
- When he turn back he travels 0.5 (Vy-Vc)+1 miles in T hours, but this time he have the current help, so his speed is Vy+Vc -> T (Vy+Vc) = 0.5 (Vy-Vc)+1 -> T Vy +T Vc = 0.5 Vy-0.5*Vc +1 -> Vy(T-0.5) + Vc(T+0.5) = 1
- The hat travels 5 Vc before the man notice it and 5 T after. This mean the hat made 1 mile in 0.5+T hours with Vc speed. -> Vc(T+0.5)=1
- From 3 and 4 we have Vy(T-0.5)=0 -> T=0.5
- From 4 and 5 we have Vc(0.5+0.5)=1 -> Vc=1 mile/hour Sorry for my english, i hope this is clear.
In number 4, did you mean ' 0.5 Vc before the man notice it and 0.5T after ' ?
Let us assume
speed of yatch = x;
speed of water current =y;
the time yatch takes to reach the dock (IN THE RETURN JOURNEY)
(t1) = { (x-y)/2 + 1 }/(x+y)
the time hat takes o reach the dock(t2) =y*1
but given that t1+1/2 =t2 (30 min=1/2 hour)
if we substitute t1 & t2 and simplify we get
x+1 =(xy) + (y^2)
on comparing y=1..
My approach too was the same, the royal approach!!
Taking the frame of reference of the ground, we have,
the absolute velocity of current = u c u r r e n t
the velocity of the of yacht in still water = v y a c h t
The time taken by the yacht from the point where the hat flew away to returning to the dock is the time taken by the hat to reach the dock. The hat flows with the same velocity as the current = u c u r r e n t , It covers 1 mile to reach the dock, hence u c u r r e n t t = 1 .
Therefore, t = v y a c h t + u c u r r e n t 3 0 ( v y a c h t − u c u r r e n t ) + 1 + 3 0 = u c u r r e n t 1
v y a c h t + u c u r r e n t 6 0 v y a c h t + 1 = u c u r r e n t 1
u c u r r e n t = 6 0 1 miles/min = 1 mile/hour
Beautiful question indeed. To say it in simple words, Hat has flown for some time with just the speed of the river. So if man with the yacht reaches the starting point at the same time as that of hat, his movement due to speed of yacht in upstream and downstream direction should cancel out each other and then only he and hat will reach the starting point at the same time. This mean the yacht moved in downstream direction for 30 minutes. Hence hat travelled 1 mile in 30 + 30 = 60 minutes. So speed of the river will be 1 mile per hour.
From the perspective of the hat, the man sailed away at the rate of his yacht for 30 minutes, and then sailed toward the hat at the same rate. This may be verified by adjusting the rates of each by the rate of the current, which results in 0 for the hat and the speed of the yacht in still water for the yacht.
So, it took the yacht 30 minutes to reach the hat after turning around.
The hat took 1 hour to travel 1 mile, and traveled at a rate of 1 mph.