# Thread: Modelling with differential equation

1. ## Modelling with differential equation

A cyclist freewheeling on a level road experiences a negative acceleration which is proportional to the square of his speed. His speed is reduced from $\displaystyle 20m/s$ to $\displaystyle 10m/s$ in a distance of $\displaystyle 100m$. Find the average speed (w.r.t. time) during this period.

Would the equation be

$\displaystyle \dfrac{dv}{dt}=-kv^2$ where $\displaystyle k$ is a constant of proportionality

2. Originally Posted by acevipa
A cyclist freewheeling on a level road experiences a negative acceleration which is proportional to the square of his speed. His speed is reduced from $\displaystyle 20m/s$ to $\displaystyle 10m/s$ in a distance of $\displaystyle 100m$. Find the average speed (w.r.t. time) during this period.

Would the equation be

$\displaystyle \dfrac{dv}{dt}=-kv^2$ where $\displaystyle k$ is a constant of proportionality
yes.

yes.
Thanks. So I'm a little unsure of what to do next.

$\displaystyle \dfrac{dv}{dt}=-kv^2$

$\displaystyle -\dfrac{dv}{v^2}=k \ dt$

$\displaystyle \displaystyle\int -\dfrac{dv}{v^2}=\int k \ dt$

$\displaystyle \dfrac{1}{v}=kt+C$

4. Originally Posted by acevipa
Thanks. So I'm a little unsure of what to do next.

$\displaystyle \dfrac{dv}{dt}=-kv^2$

$\displaystyle -\dfrac{dv}{v^2}=k \ dt$

$\displaystyle \displaystyle\int -\dfrac{dv}{v^2}=\int k \ dt$

$\displaystyle \dfrac{1}{v}=kt+C$
You have to set up a differential equation which involves the displacement and velocity.

Hint: Notice that a=dv/dx=(dv/dx) x (dx/dt) = v (dv/dx)

5. Originally Posted by acevipa
Thanks. So I'm a little unsure of what to do next.

$\displaystyle \dfrac{dv}{dt}=-kv^2$

$\displaystyle -\dfrac{dv}{v^2}=k \ dt$

$\displaystyle \displaystyle\int -\dfrac{dv}{v^2}=\int k \ dt$

$\displaystyle \dfrac{1}{v}=kt+C$
... all right!... the solution is then...

$\displaystyle \displaystyle v(t)= \frac{1}{k t + c}$ (1)

... where of course is $\displaystyle c= \frac{1}{v(0)}$. Now suppose that $\displaystyle v(0)$ is negative, for example $\displaystyle v(0) = -1$, so that the solution is...

$\displaystyle \displaystyle v(t)= \frac{1}{k t -1}$ (2)

A little question: what does if happend at the time $\displaystyle t=\frac{1}{k}$ ? ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

You have to set up a differential equation which involves the displacement and velocity.

Hint: Notice that a=dv/dx=(dv/dx) x (dx/dt) = v (dv/dx)
Thanks that makes a lot of sense.

$\displaystyle a=-kv^2 \Longrightarrow \dfrac{dv}{dt}=-kv^2$

$\displaystyle v\dfrac{dv}{dx}=-kv^2$

$\displaystyle \dfrac{dv}{dx}=-kv$

$\displaystyle \displaystyle\int\dfrac{dv}{v}=\int -k \ dx$

$\displaystyle \ln v=-kx+C$

When $\displaystyle x=0, v=20 \Longrightarrow C = \ln 20$

$\displaystyle \ln v =-kx+\ln 20$

When $\displaystyle x=100, v=10$

$\displaystyle \ln 10=-100k+\ln 20$

$\displaystyle 100k = \ln 20 - \ln 10$

$\displaystyle 100k = \ln 2$

$\displaystyle k = \dfrac{\ln 2}{100}$

Have I done everything right? If so, I'm not too sure what to do next.

The answer in the book is $\displaystyle 20\ln 2$

7. Originally Posted by acevipa
Thanks that makes a lot of sense.

$\displaystyle a=-kv^2 \Longrightarrow \dfrac{dv}{dt}=-kv^2$

$\displaystyle v\dfrac{dv}{dx}=-kv^2$

$\displaystyle \dfrac{dv}{dx}=-kv$

$\displaystyle \displaystyle\int\dfrac{dv}{v}=\int -k \ dx$

$\displaystyle \ln v=-kx+C$

When $\displaystyle x=0, v=20 \Longrightarrow C = \ln 20$

$\displaystyle \ln v =-kx+\ln 20$

When $\displaystyle x=100, v=10$

$\displaystyle \ln 10=-100k+\ln 20$

$\displaystyle 100k = \ln 20 - \ln 10$

$\displaystyle 100k = \ln 2$

$\displaystyle k = \dfrac{\ln 2}{100}$

Have I done everything right? If so, I'm not too sure what to do next.

The answer in the book is $\displaystyle 20\ln 2$
That's correct. Then, work from $\displaystyle \dfrac{dv}{dt}=-kv^2$

For velocity, integrate from 20 to 10 and for time, integrate from 0 to t.