1. ## Dynamics 3

A bullet fired from a rifle with a muzzle velocity of 1000m/s experiences a retardation due to air resistance of 0.001v^2 m/s^2 where v is the speed in m/s. What will be its speed when it has travelled a distance of 1000 metres?
(I've actually tried this question 3 times.. and 3 different approaches failed miserably! Please help me out. By saying this.. It reminds me of what Thomas Edison once said "Rather than saying that I've failed 1000 times in this project, I would say, there's a thousands different ways to fail this project" )

2. Originally Posted by Modernized

A bullet fired from a rifle with a muzzle velocity of 1000m/s experiences a retardation due to air resistance of 0.001v^2 m/s^2 where v is the speed in m/s. What will be its speed when it has travelled a distance of 1000 metres?
(I've actually tried this question 3 times.. and 3 different approaches failed miserably! Please help me out. By saying this.. It reminds me of what Thomas Edison once said "Rather than saying that I've failed 1000 times in this project, I would say, there's a thousands different ways to fail this project" )
Assume the we are operating in the regium where $v>0$, then

$\dot{v}=-0.001 v^2$

which is an ODE of variable seperable type, so:

$\int v^{-2} dv=- \int 0.001 dt$

or:

$-1/v=-0.001t-1/v_0$

rearrange to give $v$ as a function of $t$.

Now integrate again to get distance as a function of $t$ and proceed from there.

CB