Find acceleration of boat pulled into dock by winch 15' above boat's deck

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pandara

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Find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.)

A boat is pulled into a dock by means of a winch 15 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 4 feet per second. Find the acceleration of the boat when there is a total of 17 feet of rope out. (Round your answer to three decimal places.)

How to do this??


 
pandara said:
Find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.)

A boat is pulled into a dock by means of a winch 15 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 4 feet per second. Find the acceleration of the boat when there is a total of 17 feet of rope out. (Round your answer to three decimal places.)

How to do this??
Click to expand...
Where is it?
 
pandara said:
Find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.)

A boat is pulled into a dock by means of a winch 15 feet above the deck of the boat (see figure). The winch pulls in rope at a rate of 4 feet per second. Find the acceleration of the boat when there is a total of 17 feet of rope out. (Round your answer to three decimal places.)

How to do this??
Click to expand...

What have you been able to do? Is there a part you don't understand? What do you know that you might use for this problem? What have you been taught about related rates?

We can probably guess what the picture looks like; but we can't guess what specific help you need.

Perhaps you have not read our guidelines for submission, as summarized here. Following them can get quicker results.

But if you just need help getting started, I would define some variables (if they are not already defined in the picture), and write an equation relating them, which will probably involve the Pythagorean Theorem.
 
Dr.Peterson said:
What have you been able to do? Is there a part you don't understand? What do you know that you might use for this problem? What have you been taught about related rates?

We can probably guess what the picture looks like; but we can't guess what specific help you need.

Perhaps you have not read our guidelines for submission, as summarized here. Following them can get quicker results.

But if you just need help getting started, I would define some variables (if they are not already defined in the picture), and write an equation relating them, which will probably involve the Pythagorean Theorem.
Click to expand...


i thought that i sent something already but i got -17/5 for spd but i dont know how to get acceleration from that i believe that i did it wrong as well
i think that we can use tan and its derv but i don't know how
i mean derv of velocity is acceleration and derv of spd is velocity so like take a derv of -17/5? i also realize that i haven't found dy/dt maybe i have to find that? what would be that tho a velocity or ??

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pandara said:
i thought that i sent something already but i got -17/5 for spd but i dont know how to get acceleration from that i believe that i did it wrong as well
i think that we can use tan and its derv but i don't know how
i mean derv of velocity is acceleration and derv of spd is velocity so like take a derv of -17/5? i also realize that i haven't found dy/dt maybe i have to find that? what would be that tho a velocity or ??

View attachment 10266
Click to expand...
How did you arrive at that number for speed to be (-17/5)?

Why do you have negative sign in front of speed? That is troublesome!!

Please share your detailed work.

First define your Cartesian co-ordinate system. The boat will travel only in horizontal direction - where as the rope is being pulled at an angle with a speed of 4 ft/sec.

You need o derive a function of speed with y (the length of the rope) as an independent variable. Then you need to calculate the time-derivative of that (twice) to calculate acceleration.
 
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The crucial point is that you have a right triangle. The length of the hypotenuse is given in the picture as "y" and one leg is 15. Calling the other leg, the distance from the boat to the dock, "x", we have, by the Pythagorean theorem, \(\displaystyle x^2+ 15^2= x^2+ 225= y^2\). Differentiate both sides with respect to time, "t": 2x x'= 2y y'. Differentiating again, \(\displaystyle 2(x')^2+ 2x x''= 2(y')^2+ 2y y''\) or, dividing by 2, \(\displaystyle (x')^2+ xx''= (y')^2+ y y''\).

At the particular instant in question, y= 17 so \(\displaystyle x^2+ 225= 17^2= 289\). \(\displaystyle x^2= 289- 225= 64\) so x= 8. 2(8)x'= 2(17)y'. x'= (17/8)y'. We are told that the rope is being pulled in at a constant 4 ft/s so y'= 4, x'= (17/8)(4)= 17/2 and y''= 0.


\(\displaystyle (17/2)^2+ (8)x''= (4)^2+ y(0)\)

Solve that for x''.
 
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