I'm working on a physics experiment that involves a parachute made from a plastic garbage bag, and I'm trying to understand the deployment time in relation to the parachute's apex vent radius. I have derived an equation that I believe accurately describes the relationship, but I'm struggling to linearize it for further analysis.
The equation is as follows:
$t = h \sqrt{\frac{\frac{1}{2} \rho C_D \pi\left(r_{\text {total }}^2-r_{\text {apex vent}}^2\right)}{g m}}$
where:
- $t$ is the time it takes for the parachute to deploy,
- $h$ is a constant height factor,
- $\rho$ is the constant air density,
- $C_d$ is the constant drag coefficient,
- $\pi$ is the mathematical constant,
- $r_{total}$ is the total radius of the parachute canopy (constant at 0.2m),
- $r_{apex vent}$ is the radius of the apex vent (variable from 0.01m to 0.05m),
- $g$ is the constant acceleration due to gravity, and
- $m$ is the constant mass of the payload.
Given that $r_{total}$ is significantly larger than $r_{apex vent}$, I'm looking for a way to linearize this equation to make the relationship between $t$ and $r_{apex vent}$ more explicit and easier to analyze.
I know all the constants can be grouped to give the equation: $t=k \sqrt{\left(c-r_{\text {apexvent }}^2\right)}$
Could anyone offer some insight or suggest a method for linearizing this equation? I'm especially interested in how the changes in the apex vent radius impact the deployment time.
Thank you for your time and help!