Crossing a river with a current (optimization)

In summary, a man is trying to cross a river that is 50 m wide and has a current flowing at 2.4 m/s. His swimming speed is 3 m/s and his walking speed is 10 m/s. To have the fastest time to cross the river, he should angle himself so that he swims at the most efficient angle.
  • #1
TheAstroMan
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1. A man is on one side of a river that is 50 m wide. He is trying to get to someone directly on the other side. There is a current flowing down the stream at 2.4 m/s. His swimming speed is 3 m/s and his walking speed is 10 m/s .

What is the best angle for him to swim at to have the fastest time to cross the river?

I've tried looking for this and I've only seen questions where the person is not directly across, and my math isn't very strong so I'm not able to use those to answer the one I have :(

Thanks!

2. 50/(3cos(x))+(2.4-3sin(x))*(50/(3cos(x)))/10 ^^ I've gotten here, which is the time to get across + the distance from the other person divided by 10 but I don't understand the calculus you have to use after this equation.

3. I just plugged that equation into wolfram alpha's minimum calculator and got two numbers. I used the one without a variable and assuming it was differentiated I assumed it was in radians. So I divided that by Pi / 180 to convert it into degrees and somehow I got the answer.

Can someone please explain the whole minimum / differentiating process? Thanks :D
 
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  • #2
what does his walking speed have to do with anything.. its not like theirs a bridge to cross so he should be just swimming right?
 
  • #3
The current speed pushes him away from the person so when he's on shore he has to walk the remainder.
 
  • #4
so how much shore is there? like 10m is shore 30m is river, 10m is shore?
 
  • #5
Nono, imagine the river is a rectangle and the width is 50. So let's say he swims straight and the current pushes him down the shore 40m . So he has now crossed the river but still needs to walk the remaining 120.

Thats what the walking speed is for :p
 
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  • #6
TheAstroMan said:
What is the best angle for him to swim at to have the fastest time to cross the river?
It says to find the most efficient route there but
TheAstroMan said:
Nono, imagine the river is a rectangle and the width is 50. So let's say he swims straight and the current pushes him down the shore 120m . So he has now crossed the river but still needs to walk the remaining 120.

Thats what the walking speed is for :p
if walking the question would be easier so his walking speed is 10m/s which (distance to walk)120/10(meters per seccond) = 12 seconds to walk up to the destination

for your question it would take (50meters ÷ 3m/s = 16.66~ seconds to cross which in that time would bring him down the river 2.4 x 16.66 = 40m down the river (roughly) so then 4 seconds to walk up to the destination

My math isn't on edge but somebody should know how to find the angle to get to the end most efficiently but i can't be bothered
 
  • #7
Alright thanks. I messed up the math, its approximately 40m down not 120 haha oops.

But yea, I think I need someone who knows calculus for this :D
 
  • #8
It was exactly 40m down cause 50÷3m/s is 16.666666666666666666666666666667 Times 2.4 = exactly 40
 
  • #9
I got here

50/(3cos(x))+(2.4-3sin(x))*(50/(3cos(x)))/10 ^^ Basically the time it takes him to get across + his distance from the other person divided by 10. (I think this is the right equation, not sure though)

Now I just need someone to explain how I can get a value for x that would give me the smallest answer :p
 

Related to Crossing a river with a current (optimization)

1. How can I determine the optimal angle to cross a river with a current?

The optimal angle for crossing a river with a current is determined by considering the direction and strength of the current, as well as the speed and stability of the person or object attempting to cross. Generally, a diagonal angle that is slightly upstream is most efficient, as it allows for the current to assist in crossing while also minimizing the distance traveled.

2. What factors should be taken into account when optimizing a river crossing?

Factors that should be considered when optimizing a river crossing include the speed and direction of the current, the speed and stability of the person or object attempting to cross, the width of the river, and the terrain on either side of the river. Other factors such as weather conditions and potential hazards in the water should also be taken into account.

3. How can I calculate the time it will take to cross a river with a current?

The time it takes to cross a river with a current can be calculated by dividing the distance to be crossed by the speed of the current. This will give an estimated time for crossing, but it is important to also consider the speed and stability of the person or object attempting to cross when determining the overall time it will take.

4. Is there a specific technique for crossing a river with a current?

There are various techniques that can be used for crossing a river with a current, depending on the specific situation. Some common techniques include using a ferrying method, where the person or object moves diagonally upstream while crossing, and using a bracing method, where the person or object leans into the current to maintain stability. The best technique will depend on the individual's abilities and the conditions of the river.

5. How can I prepare for a river crossing with a current?

It is important to properly prepare for a river crossing with a current by assessing the conditions beforehand, including the speed and direction of the current, the width of the river, and any potential hazards. It is also important to have the necessary equipment, such as a personal flotation device, and to be physically and mentally prepared for the crossing. Having a plan and being aware of the potential risks can help ensure a successful and safe river crossing.

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