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Dynamics problem

  • Thread starter rammer
  • Start date
  • #1
23
0

Homework Statement



A boat has mass m=1000 kg and is moving at velocity v0=324 ms^-1. Friction force btw the boat and water is proportional to velocity v, Fd=70*v. How long it takes to slow down to 162ms^-1 ?


Homework Equations


I'm not sure which function should I integrate, because acceleration is not constant.


The Attempt at a Solution


I understand Friction force and acceleration as functions of v, but I have no idea how to express these as functions of time, since acc is not constant. Then I would integrate a(t) with respect to time and substitute final velocity for v(t) and from that I'd get the answer.
 

Answers and Replies

  • #2
327
2
Draw a force diagram, and apply Newton's 2nd law. You can get the acceleration that way. Then use [itex]a = \frac{dv}{dt}[/itex] to get a differential equation. Solve it and find [itex]v[/itex] in terms of [itex]t[/itex]. Solve for [itex]t[/itex] and substitute correct value of [itex]v[/itex] to find the time.
 
  • #3
23
0
a=-70v/m
∫dv=∫(-70v/m)dt
v=-70/m ∫vdt - the problem is I do not know v in terms of t :(
 
  • #4
327
2
Instead of your second line, do this,
[tex]\int{\frac{dv}{v}}=\int{\frac{-70 dt}{m}}[/tex]
 
  • #5
23
0
I have to correct given informtion v0=25 ms^-1 and v=12.5 ms^-1

Ok, I tried your suggestion and from that I get:

t = (-m*(ln(v) + v0))/70 -what doesn't seem right

After substituting v = 12.5 I get t = 393 s and that is wrong (correct answer should be 9.9s)
 
  • #6
327
2
t = (-m*(ln(v) + v0))/70
This does not agree with my final answer.

Check whether you applied the initial boundary condition (v = v0 when t=0 ) correctly.
 
  • #7
23
0
At t = 0, I'm pretty sure, the integration constant is equal vo (=25).
 
  • #8
327
2
Not ln(vo) ?
 
  • #9
23
0
Yes, you're right, thanks. I finally got it correct. My mistake was I put the constant directly from initial conditions, not solving from integrated function.
 

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