Finding Time for Boat to Slow Down with Velocity Dependent Forces

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To determine the time required for a 1000 kg boat to slow from 25 m/s to 12.5 m/s after shutting off its engine, the frictional force is given as f = 70v, where v is the speed. The acceleration can be expressed as a(t) = -70v(t) / 1000 kg, indicating that the force is acting in the opposite direction of motion. The student struggles with finding the anti-derivative of acceleration since it is defined in terms of velocity, and they cannot apply the kinematic equation without knowing the distance. They plan to seek clarification from their teacher regarding the calculus concepts involved. Understanding the relationship between velocity and acceleration is crucial for solving this problem.
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Homework Statement


A 1000kg boat is traveling at 25 m/s when its engine is shut off. The magnitude of the frictional force f between the boat and the water is proportional to the speed v of the boat: f=70v, where v is in meters per second and f is in Newtons. Find the time required for the boat to slow to 12.5 m/s.


Homework Equations


F=ma
a(t)=v'(t)

The Attempt at a Solution


70v = (1000 kg) * a
a(t) = 70v(t) / (1000 kg)

However, after this point I don't know what to do. I can't use the formula v2 = vo2 + 2a(Δx), because I don't know Δx. I don't know how I would find the anti-derivative of a(t), since a(t) is defined in terms of v(t). What am I missing?
 
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Does this look right?
 

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It probably is, but I don't understand it. We haven't gotten that far in ap calculus yet, so I'll ask my teacher about how that works tomorrow. Thanks for the help!
 

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