Optimizing Swim and Run Angle for Crossing a River

[vthenry]

Homework Statement

find the optimal angle so that the time required for the swimmer,runner to cross the river (directly opposite starting position) is minimum. the swimmer swims to one point at the end of the river then he/she will run to the target(opposite from starting position)

swim speed 4miles/hour
running speed 10miles/hour
river flow speed 2miles/hour

this is all the information that is given

2. The attempt at a solution

http://img13.imageshack.us/img13/2500/extracredit.jpg

h=width of river
k=distance for running part of problem
z=distance covered by the resultant velocity
$$\theta$$= angle swimmer takes off
V_swim=velocity swimmer swims
V_run=velocity he/she run
V_river=velocity of river flow
V_res=resultant velocity of river flow and swimming

ok so here is what i have so far,

V_res²=[V_swim*sin(90-$$\theta$$)]²+[V_river-V_swim*(cos (90-$$\theta$$)]²

therefore:

V_res²=[V_swim*cos$$\theta$$]²+[V_river-V_swim*sin$$\theta$$]²


it is also true from diagram that
z²=h²+k²

now:

equation 1 and 2

z²=[V_res*t]²=t²*V_res²

k=V_run*t


ok...

it is also true that

equation 3:
z²=h²+k²

sub equation 1 and 2 into equation 3


... here is where i am stuck...

what do i do now
i can only think of differentiating d(theta)/d(t) and set that to zero... is that even correct i have no idea where to go on from this even when i try d(theta)/d(t) and i differentiate the equation i am still left with the varible time...


this is an interesting problem. and i would like to know how to solve it

any help would be appreciated , thank you
vthenry

 

[LowlyPion]

Welcome to PF.

Since time is the variable that needs to be optimized, I'd suggest working in equations that represent the times of the segments involved. Then you can optimize that function in t, to find its minimum as you suggest by taking the derivative.

For instance Time to swim across will be the distance h divided by his y component of velocity. Then figure his time to run the distance k along the bank as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land. (If he swims at an angle with the current then this will necessarily be longer so you can discard that as an option if you want his minimum time.)

 

[vthenry]

hey thanks for helping =)

ill try it and let you know how i do

cheers,
vthenry

 

[vthenry]

Welcome to PF.

Since time is the variable that needs to be optimized, I'd suggest working in equations that represent the times of the segments involved. Then you can optimize that function in t, to find its minimum as you suggest by taking the derivative.

For instance Time to swim across will be the distance h divided by his y component of velocity. Then figure his time to run the distance k along the bank as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land. (If he swims at an angle with the current then this will necessarily be longer so you can discard that as an option if you want his minimum time.)

hey i don't understand this part
"as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land"

thanks

 

[LowlyPion]

hey i don't understand this part
"as that time to cross times the river speed less whatever angle against the current he swam and all that divided by his running velocity on land"

thanks

The distance k is determined by the speed the swimmer is carried down the river. The first term. The second term takes account of the angular component of his velocity that is || to the direction of the current. That may be either a + or -, but since you are wanting to minimize the time you would take (-) that component times the time right?

The distance k/Vrunning is the time it takes to run along the other bank.

 

[vthenry]

ok so, i have this equation now which i have to take the derivative of

T_total = k/Vrun + h/(Vswim(y component))

k=T_swim(Vriver-Vswim(x component))

 

[LowlyPion]

ok so, i have this equation now which i have to take the derivative of

T_total = k/Vrun + h/(Vswim(y component))

k=T_swim(Vriver-Vswim(x component))

That looks about right.

Now you should have T_(θ), so you need to figure the minimum for that function.

 

[vthenry]

That looks about right.

Now you should have T_(θ), so you need to figure the minimum for that function.

wow the equation i have to solve is too difficult
set dt/d(theta) = 0 and i have this eqaution T(theta)

T=t_swim+t_run

T=$$\frac{h}{Vswim*cos\theta}$$ + ($$\frac{Vriver-Vswim*sin\theta}{Vrun}$$)*t_swim

how am i going to solve this now for the optimal angle to minimize time ?

 

[LowlyPion]

wow the equation i have to solve is too difficult
set dt/d(theta) = 0 and i have this eqaution T(theta)

T=t_swim+t_run

T=$$\frac{h}{Vswim*cos\theta}$$ + ($$\frac{Vriver-Vswim*sin\theta}{Vrun}$$)*t_swim

how am i going to solve this now for the optimal angle to minimize time ?

It looks to me that you can differentiate this with respect to θ and set it to 0 to determine the minimum of the function T_(θ)

Sub in the values given in the problem and rewrite the terms in terms of Sec and Tangent and then differentiate right?

 

[LowlyPion]

Your equation should likely be simplified further by the way to

T = h/(Vsm*cosθ ) * (1 + (Vrvr - Vsw*sinθ)/Vrun )

 

[vthenry]

Your equation should likely be simplified further by the way to

T = h/(Vsm*cosθ ) * (1 + (Vrvr - Vsw*sinθ)/Vrun )

ok somehow after differentiating and setting d(t)/d(theta) = 0

and some algebraic minipulation i am left with this

cos(theta)+sin(theta)tan(theta)+tan(theta)=0

i have no idea if this is correct or not.. but how do i solve this now ?

 

[LowlyPion]

ok somehow after differentiating and setting d(t)/d(theta) = 0

and some algebraic minipulation i am left with this

cos(theta)+sin(theta)tan(theta)+tan(theta)=0

i have no idea if this is correct or not.. but how do i solve this now ?

I converted my equation to

T = h/4*(secθ) + 2h/40*secθ - 4*h/40*tanθ

That yields

T = 12*h*secθ/40 - h*tanθ /10

Now just differentiate that and set it to 0.