Finding the function

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December 15th 2006, 04:28 AM
Jones

Finding the function

Hello,

I need help with this one.

"During a boat trip, the engine suddenly stops. If the breaking force is proportional to the velocity it is possible to set up a differential equation $m*dv/dt=c*v$ where m is the mass of the boat, v is the velocity, t is the time and c is the braking force"

a) Determine v as a function of time if t=0 yields v= $v_0$

How am i supposed to do this?
December 15th 2006, 08:12 AM
Soroban

Hello, Jones!

Quote:

During a boat trip, the engine suddenly stops.

If the braking force is proportional to the velocity,
it is possible to set up a differential equation: $m\cdot\frac{dv}{dt}\:=\:kv$
where $m$ is the mass of the boat, $v$ is the velocity,
$t$ is the time and $k$ is the braking force.

a) Determine $v$ as a function of time if $t=0$ yields $v = v_0$

We are given: . $m\cdot\frac{dv}{dt} \:=\:kv$

Separate variables: . $\frac{dv}{v}\:=\:\frac{k}{m}dt$

Integrate: . $\ln v \:=\:\frac{k}{m}t + c$

. . and we have: . $v \:=\:e^{\frac{k}{m}t + c} \:=\:e^{\frac{k}{m}t}\cdot e^c \:=\:e^{\frac{k}{m}t}\cdot C$

Hence: . $v(t)\:=\:Ce^{\frac{k}{m}t}$

We are told that: . $v(0) = v_o$
. . So we have: . $v_o \:=\:Ce^{\frac{k}{m}(0)}\quad\Rightarrow\quad C = v_o$

Therefore: . $\boxed{v(t)\;=\;v_oe^{\frac{k}{m}t}}$
December 15th 2006, 09:33 AM
Jones

Im i supposed to know this stuff? :(

Thanks for the help.
December 15th 2006, 01:21 PM
Jones

Quote:

Originally Posted by Soroban

Hello, Jones!

[size=3]
We are given: . $m\cdot\frac{dv}{dt} \:=\:kv$

Separate variables: . $\frac{dv}{v}\:=\:\frac{k}{m}dt$

Isn't $dt$ just with respect to the variable you are differentiating?

Why are we substituting for $dt$ ?
December 16th 2006, 04:15 AM
topsquark

Quote:

Originally Posted by Jones

Im i supposed to know this stuff? :(

Thanks for the help.

Quote:

Originally Posted by Jones

Isn't $dt$ just with respect to the variable you are differentiating?

Why are we substituting for $dt$ ?

All Soroban is doing is the "separation of variables" method of solving a differential equation.

-Dan