Newton’s second law

**mikegar813** · September 22nd 2009, 10:27 AM

Suppose an object traveling through a fluid is subject only to a force resisting its motion, and this resisting force is proportional to the velocity of the object. By Newton’s second law we have m(dv/dt)=-cv. Find a solution v(t) of this differential equation for which v(0) = v sub 0.

**skeeter** · September 22nd 2009, 10:41 AM

Originally Posted by mikegar813

Suppose an object traveling through a fluid is subject only to a force resisting its motion, and this resisting force is proportional to the velocity of the object. By Newton’s second law we have m(dv/dt)=-cv. Find a solution v(t) of this differential equation for which v(0) = v0.

$ m \frac{dv}{dt} = -kv $

$ \frac{dv}{dt} = -\frac{k}{m}v $

rate of change of velocity is directly proportional to itself ...

$ v = v_0 \, e^{-\frac{k}{m}t} $

**mikegar813** · September 22nd 2009, 11:10 AM

I don't even know where to begin with this one.

**HallsofIvy** · September 22nd 2009, 02:07 PM

As Skeeter said, you have $\frac{dv}{dt}= -\frac{k}{m}v$ .

Separate that into "differentials": $\frac{dv}{v}= -\frac{k}{m}dt$ .

Now integrate both sides and then solve for v.

**mikegar813** · September 23rd 2009, 03:27 AM

ok thanks

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