How can I set up an equation for distance traveled for a rubber band powered car? I am to make a rubber band powered car, and I have to maximize it's distance traveled. The time taken to reach said distance, top speed of the car, acceleration of the car, is not relevant. I just want to maximize distance.
What would be the easiest way to go about this? I am thinking I could either take the route of Torque ($T=Fr=Ia$), where I would break up inertia into the axle and both wheels on each side. Or I could take the route of energy ($\frac 1 2 kx^2 = \frac 1 2 mv^2 = \text{work}$?).
I'm very confused by this as there are so many things involved. Any clarification and help would be appreciated.
 A: You want to consider the following factors:


*

*Maximize initial energy stored in the rubber band. This means you need to be able to twist the band lots of times, and as it unwinds it must continue to produce torque.

*Minimize internal friction - make the mechanism that converts power from the band to kinetic energy as "direct" as possible. In fact a rubber band untwisting can lose a lot of energy; so would any gearing mechanism

*Minimize air drag: drag goes as $$F = \frac12 \rho A C_D v^2$$ where $\rho$ is the air density (about 1.2 kg/m3), $A$ is the apparent area (looking at the front of the car), $C_D$ is the drag coefficient (about 0.5 for a sphere, smaller for a more aerodynamic car), and $v$ is the velocity. This is why a lower average velocity will help you - but how do you persuade the car to move slowly without a gearing mechanism... (see point 2 above!)

*Minimize road friction. Rubber tires (especially not well inflated ones) will add to drag; hard wheels work better (which is why trains are so efficient), and large wheels work better (they ride over the surface without "digging a hole" for themselves).


Now in order to keep the maximum velocity low, you want a car that is as heavy as possible, with large low-friction wheels. Being heavy, it will prevent the car from initially going very fast - while storing all the energy from the elastic band as low velocity kinetic energy; this keeps the air drag down. But you need excellent wheels and a hard smooth surface (point 4) in order to make it so the heavy car doesn't suffer from rolling friction.
So you need to experiment. If the rolling friction can be modeled as $F_r = m g \mu_1$ and the air friction as $F_d = \alpha (\frac12 m v^2)/m$, you can see that for the same kinetic energy $\frac12 m v^2$ you will get lower air drag but higher rolling friction as you increase $m$. 
This means that if you play around with the mass, you will find there is an optimum: either very light, or very heavy, cars will not work as well as the ones that are "just right".
Goldilocks. Every time.
