1. ## boat crossing river

the velocity of a boat relative to the water is $5i-2j m/s$. the current has a speed of 5 m/s. if the boat as a result moves in a North east direction find two values of the (i)velocity of the current and (ii) the resultant velocity of the boat

I am confused on if the boat actually moves in the direction of the boat relative to the current velocity or the speed and direction at which the boat sets out

I know velocity of boat relative to current =velocity of boat -velocity of current

2. ## Re: boat crossing river

boat vector relative to the water + current vector = boat vector resultant

$(5i-2j) + (ai + bj) = (di + dj)$

$5+a=d \implies a > -5 \text{ since } d > 0$ why?

$b-2=d \implies b> 2$

$5+a=b-2 \implies b-a = 7$

choose $b=3 \implies a=-4 \implies d=1$

current vector = $-4i+3j$, boat resultant = $i + j$

choose another value for $b$ that fits the restriction and work out the corresponding value for $a$

3. ## Re: boat crossing river

Would there not be a better way to solve this than guess work?
I should of mentioned the part a of the question was given that sin^2 (a) +cos^2 (a)=1 show cos (a) in terms of sin (a) which is cos (a)=✓(1-sin^2 a).

So you probably have to use that in the question but I can not see where
I am sorry for not using latex but I am on my phone and it is difficult to do so.

4. ## Re: boat crossing river

Originally Posted by markosheehan
Would there not be a better way to solve this than guess work?
I should of mentioned the part a of the question was given that sin^2 (a) +cos^2 (a)=1 show cos (a) in terms of sin (a) which is cos (a)=✓(1-sin^2 a).

So you probably have to use that in the question but I can not see where
I am sorry for not using latex but I am on my phone and it is difficult to do so.
Part (a), as you call it, is just proving a trig identity (which is incorrect as you've posted it) ...

$\sin^2{a} + \cos^2{a} = 1 \implies \cos^2{a} = 1 - \sin^2{a} \implies |\cos{a}| = \sqrt{1-\sin^2{a}}$

What this has to do with the boat vector problem is beyond me ... maybe you should post the entire problem?

btw, it's not "guess work" .... there are many valid solutions to the boat vector problem. The question you posted asked for two values for the current and resultant vectors. The attached diagram shows five of the many possible solutions based upon the information you provided.

5. ## Re: boat crossing river

I have posted the whole problem. I understand your method. thanks skeeter as usual

6. ## Re: boat crossing river

I know the other way to solve it. so
b=boat and c=current
Vbc = 5i-2j

Vc= 5cosai + 5sinaj

Vbc=Vb-Vc

Vb= (5+5cosa)i + (-2+5sina)j

this is in a north east direction

$tan 45= (-2+5sina)/(5+5cosa)$

we know | cos (a)|=✓(1-sin^2 a) so we can sub this in.

from here you solve it like a quadratic equation.

solving this gives cosa=4/5 or cosa=3/5