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posted by  wvlilgurl on 10/6/2008 3:00:24 PM  |  status: Live  

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A motorboat is moving at 30m/s when its motor suddenly quits and it is 800 m from the shore.   Ten seconds later the boat has slowed to 20 m/s.  Assume that the resistance it encounters while coasting  is proportional to its velocity.  How far will the boat coast in all?  Can the boat reach the shore?
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posted by toadatrix on 10/11/2008 12:59:12 AM  |  status: Live
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If we let d[t] represent the distance the boat has traveled toward the shore after t seconds the differential equation of motion would be:
 
         d''[t]==-k d'[t], where k is a constant to be determined
 
The initial conditions are d[0]=0 and d'[0]=30.
 
Solve the differential equation (I assume you can do that) and you obtain:
 
         d[t] = (30 - 30/E^(k*t))/k, where E is the transcendental number e. Plug in the value of t=10 and solve for k such that
                                                   d'[10] = 20 and you find that k = (ln(3/2))/10. So we then have:
 
          d[t] = (-300*(-1 + (2/3)^(t/10)))/ln[3/2]
 
To find how far the boat goes take the limit of d[t] as t->Infinity and you obtain 300/ln[3/2] = 739.89. So the boat will never reach the shore.
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