Fluid Friction Question

1. Sep 23, 2011

Dragon M.

1. The problem statement, all variables and given/known data

I'm trying to compare the velocities at 5 meters of three projectiles subject to air resistance: the first with an initial velocity of 121.632 m/s, the second with an initial velocity of 136.8m/s, and the third at 182.442 m/s.

All three projectiles have a mass of 2.0x10^-4 kg (m), a cross sectional area of 2.81x10^-5 m^2 (A), and drag coefficient of .47 (Cd). Density of air is assumed to be 1.204 kg/m^3.

2. Relevant equations

The equation I have been primarily been using is the drag equation Fa = 0.5Dv^2CdA

My initial plan was to use F = ma and Vf^2 = Vi^2 + 2a$\Delta$x. However, I realized after doing these calculations that Fa changes with respect to velocity.

3. The attempt at a solution
This is my attempt to the solution of finding the velocity after 5 meters for 121.632 m/s.

Fa = -(0.5)(1.204)(121.632 m/s)^2(.47)(2.81E-5) = -1.2E-1

Which I then realized that only applied initially at launch.

So I tried taking the derivative with respect to time.

dFa = (.47)(2.81E-5)(dv/dt)

I am fairly stuck at the moment. Where do I go from now? If there is not enough information, what information do I need and what hints would you give to experimentally gain this information?

2. Sep 23, 2011

omoplata

Here is my suggestion.

Using the drag equation, Newton's second law and the fact that $a = \frac{d v}{d t}$ (where $a$ is the acceleration) you can find a differential equation for $v$. Solving that, you can find $v$ at any time $t$. Then, using the fact that $v = \frac{d x}{d t}$, you can find another differential equation for $x$. Solving this, you can find the time at $x = 5m$. Then you can plug in that time to the equation for $v$.

Let me know if it works.