# Newton’s second law

• September 22nd 2009, 10:27 AM
mikegar813
Newton’s second law
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.
• September 22nd 2009, 10:41 AM
skeeter
Quote:

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}
$
• September 22nd 2009, 11:10 AM
mikegar813
I don't even know where to begin with this one.
• September 22nd 2009, 02:07 PM
HallsofIvy
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.
• September 23rd 2009, 03:27 AM
mikegar813
ok thanks