# Approximating Gallons of Gas Needed for Trip (read: physics grad. with too much free time)

My departure from college has left me with an itch for physics-y problems to solve, and my boredom and free time gave way to this approximation of the gallons of gas I use for the trip to my girlfriends house.

1. First approximated miles per gallon (mpg) as a function of velocity [mph]

$$\boxed{mpg(v)}$$: (Function approximated from this image from the Wikipedia page on Fuel Economy) 1. Then approximated the speed (mph) as a function of time (hrs) for the trip. (n.b. this graph assumed no traffic, and is therefor horribly inaccurate as I live in LA, and there's literally always traffic in LA)

$$\boxed{v(t)}$$: 1. Checked the approximation by integrating $$v(t) \ dt$$ over the trip and checking that the total distance travelled was right (it was :D)

$$\boxed{x(t)}$$: 1. Then used $$mpg(v)$$ and $$v(t)$$ to get $$mpg(v(t))=mpg(t)$$.

$$\boxed{mpg(t)}$$: 1. The reciprocal of miles per gallon is gallons per mile, i.e. $$\dfrac{1}{mpg(t)}=gpm(t)$$, so I assumed that gallons per mile times miles per hour would give gallons per hour as a function of time $$\dfrac{v(t)}{mpg(t)}=gph(t)$$

$$\boxed{gph(t)}$$: 1. Then integrated $$gph(t) \ dt$$ over the whole trip to get gallons used: However, this seems quite low. From personal experience, I'd say I use closer to 2 gallons for this trip. I'd say my assumption of no traffic probably had some effect, but even with that correction, it's still only around 1.5.

Did I make any glaringly false assumptions? Any way I could make this approx. more accurate?

Also, for anyone who knows cars, why does it seem like I use gas more quickly when my gas tank is less full than when its more full? One would think that less gas=less weight=better fuel economy... Something to do with the shape of the tank?

• Mpg(v) is steady state? What about the cost of acceleration? – Jim Garrison Dec 6 '15 at 3:05
• "why does it seem like I use gas more quickly when my gas tank is less full than when its more full? One would think that less gas=less weight=better fuel economy... Something to do with the shape of the tank?" Probably because the gauge is not a linear function of the volume of remaining fuel. – DanielSank Dec 6 '15 at 7:15