Related Rates...Please Help

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doctorgk

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A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat.

A-._
-..... _
- ........_
- ..........._
- ............._
- ................_
C-................ _B

Thats a 90 degree right triangle, the boat is at B, the winch at A, and line AC is the 12 feet.
The winch pulls in rope at a rate of 4 feet per second. there is 13 feet of rope out (line AB), find the acceleration of the boat.


okay, so i solved for line CB using the pythagorean theorem, which is 5 feet. i know i have to implicitely differentiate, then a second time because it is acceleration. could anyone help me. i know that dx/dt is 10.4 feet/sec
 
If the boat is moving at a velocity of 10.4 feet/sec, what does that tell you about its acceleration?
 
well it tells me that the acceleration is constant because there is no change in velocity at that point, but because that velocity is a dx/dt, there has to be a change in velocity, hence an acceleration
 
If the winch is pulling the boat at a constant rate, then the boat must be moving at a constant rate. The only way for the boat to be accelerating is if the winch was accelerating but it's not. It's being pulled at a rate, as given, of 4 feet per second.
 
Since this is linear motion at a constant speed - the acceleration is zero.

The linear part is important to remember - because if the boat was pulled up on a curved ramp (at a constant rate) - it would have an acceleration.
 
This appears to be a related rates where they want dx/dt given D=13, y=12, x=5, dD/dt=4

\(\displaystyle D^{2}+x^{2}+y^{2}\)

By Pythagoras, \(\displaystyle x=\sqrt{13^{2}-12^{2}}=5\)

Differentiate:

\(\displaystyle D\frac{dD}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}\)

\(\displaystyle 13(4)=(5)\frac{dx}{dt}+0\)

\(\displaystyle \frac{dx}{dt}=\frac{52}{5}\)

Same as o_O.
 
I fully agree with o_O.

However, why are you calculating dx/dt?

The acceleration is zero - and that is the only thing what the question asked for (as posted).

I was just pointing out that for curvelinear motions - the acceleration is non-zero - even if the speed (ds/dt) is zero.
 
Mmm why would a ramp cause the boat to accelerate unless you take into account of the forces of gravity and friction ...

Edit: Oh sorry, didn't see that you wrote 'curved' implying centripetal motion.
 
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