Rate of change of a right triangle

[vande060]

Imagine a boat on the water being putted
towards the shore by a winch mounted a height
h above the water. The winch reels in cable
(shortening the hypotenuse) at constant rate
u in m/s . When the boat is a distance d from
the shore, find its speed and acceleration.
This is an example of motion at non-uniform
acceleration.
im letting hypotenuse here = x, so x^2 = d^2 + h^2
Im thinking i want to solve for d' to get the velocity at of d at distance d. okay I am going to give it a shot, but bear with me, i have little confidence in this answer and i understand that reading these next steps will probably be a little frightening for those who know how to solve this.
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x^2 = d^2 + h^2

d^2 = x^2 - h^2

2dd' = 2xx' - 2hh'

d' = [2xx' - 2hh']/(2d) <--- this is what i was thinking to explain the speed of d at distance d . my intuition was the to allow x' be u to explain the constant rate of the shortening of the hypotenuse, but I am not sure about it. i want to differentiate again to find d'' to explain acceleration, but i need to establish the correct way to do this first step before moving on.

can anyone give me a pointer or two.

 

[rl.bhat]

Whether the height h is variable or constant?

 

[vande060]

it is fixed, i believe. this is a copy of the illustration in my book:

http://s861.photobucket.com/albums/ab174/alkaline262/?action=view&current=physicstriangle.jpg

 

[rl.bhat]

it is fixed, i believe. this is a copy of the illustration in my book:

http://s861.photobucket.com/albums/ab174/alkaline262/?action=view&current=physicstriangle.jpg

In that case the derivative of h is equal to zero. And the equation becomes

2d*d' = 2x*x'

d' = x*x'/d...(1)

d'*d' + d*d" = x'*x'...(2) because x' is constant.

Now proceed.

 

[vande060]

In that case the derivative of h is equal to zero. And the equation becomes

2d*d' = 2x*x'

d' = x*x'/d...(1)

d'*d' + d*d" = x'*x'...(2) because x' is constant.

Now proceed.

okay, i understand how you got 2, you multiplied both sides by d then differentiated right?

okay so now i just simplify..

(x'*x' - d'*d')/d = d"

x' here represent the constant rate, u m/s, that the hypotenuse is shortening.

 

[rl.bhat]

Yes.

 

[vande060]

Yes.

thank you greatly