Fluid Friction Question: Calculating Projectile Velocities with Air Resistance

[Dragon M.]

Homework Statement

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 (C_d). Density of air is assumed to be 1.204 kg/m^3.

Homework Equations

The equation I have been primarily been using is the drag equation F_a = 0.5Dv^2C_dA

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

The Attempt at a Solution

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

F_a = -(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.

dF_a = (.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?

 

[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.