Finding Time for Boat to Slow Down with Velocity Dependent Forces

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SUMMARY

The discussion focuses on calculating the time required for a 1000 kg boat, initially traveling at 25 m/s, to decelerate to 12.5 m/s after its engine is turned off. The frictional force acting on the boat is defined as f = 70v, where v is the speed in m/s. The user correctly identifies the relationship between force, mass, and acceleration using F = ma, but struggles with integrating acceleration as it is expressed in terms of velocity. The solution requires applying calculus concepts to relate velocity and time, specifically through the use of differential equations.

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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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