Why Should the Boat Head in a Specific Direction to Cross the River Quickly?

[mikeyman2010]

Hey, I'm a high school physics student and I came across this question on one of my practice papers that stumbled me quite a bit. It's a relatively simple relativity/vector question, but I can't understand why the correct answer is what it is. Can any1 help me?

Here's the question:

Suppose we have a river, and a boat is trying to get to the other shore. A current is blowing to the east. In what direction should the boat head to arrive at the other end of the river in the shortest time?
a) North
b) Northeast
c) Northwest


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[AKG]

Does it matter where it ends up on the other side, i.e. does it have to end up directly across from it's starting position? If it has to be directly across, then it's only choice is to go Northwest. If it just has to get to the other side, then "North" is the correct answer. Why? Because the effect of the current only affects how fast it goes east/west, it has no north component. So, the speed at which it goes north depends solely on the north component of the rowing speed (the speed of the boat if it were in still water). Since going straight north gives the greatest possible north component, north is your answer.

 

[mikeyman2010]

Thnx for the reply. I don't think where the boat lands matters, but i'd like to know, if you go north directly, wouldn't the distance you need to travel to reach the opposite shore increase since current blows you off-course? (Since it's a hypotenuse side of a right triangle). I thought it was going northwest would give you the shortest time since it minimizes the distance you need to travel by allowing the current to blow you into a straight path. Can you explain why having a greater velocity matters more than having a shorter distance to travel to travel across the river? Again, thnx for the help!

 

[Doc Al]

I don't think where the boat lands matters, but i'd like to know, if you go north directly, wouldn't the distance you need to travel to reach the opposite shore increase since current blows you off-course? (Since it's a hypotenuse side of a right triangle).

Sure you'll travel a greater distance, but who cares? All you care about is traveling as fast as possible in the North direction. You want to minimize the time it takes to get you across the river, not the total distance you travel.

I thought it was going northwest would give you the shortest time since it minimizes the distance you need to travel by allowing the current to blow you into a straight path.

It does minimize the distance, but you waste speed going sideways. So it ends up taking you longer to cross.

Can you explain why having a greater velocity matters more than having a shorter distance to travel to travel across the river?

The only distance that matters is the distance across the river (south to north), not the total distance traveled. That south to north distance doesn't change. And you want to traverse that distance as fast as possible. So don't waste any speed fighting the current. Make sense?

 

[mikeyman2010]

Oh, i see, so even if you go straight, and get blown off-course, the distance you travel from south to north stays the same, right?

 

[Doc Al]

Exactly right!