# Escaping Road Rage

## Basic Approach

Use the work/energy method for figuring out the range of a vehicle.

Take the derivative of the range function with respect to velocity, set that equal to zero.

Solve for velocity.

## Assumptions

The model I used makes several assumptions.

- The conversion of gasoline chemical energy to work produced by the engine is perfect. If we knew the efficiency, we could simply multiply in that factor.
- The engine has no power limits and can produce infinite power at the price of using infinite gasoline.

The second assumption isn't as bad as it sounds. It just means that we might have to drop our results to the upper limit of the engine's horse power.

## Solution

Let's begin! The force required to keep a vehicle in motion has two parts--aerodynamic resistance and rolling resistance.

F=(1/2)C_{d}Aρv^{2} + μ_{k}N

c_{d} - coefficient of drag of vehicle

A - area in the direction of the velocity (frontal area)
v - Velocity of vehicle (speed)
ρ - density of air (1.225 kg/m^{3} at sea-lever)

μ_{k} - Coefficient of rolling resistance (often assumed to be 0.015 for most vehicles)

N - Normal force (equal to weight of vehicle)

We also know that work is force times a distance.

W = f*d

The ultimate goal here is to be able to drive a greater distance than your pursuer, so we solve the work equation for distance.

d = W/f

We already know the required force, so we can substitute that in!

d = W / [(1/2)C_{d}Aρv^{2} + μ_{k}N] ...simplify:

d = 2W / [C_{d}Aρv^{2} + 2μ_{k}N]

Excellent! We now have a function that can tell us the distance a vehicle can travel if we know how much work it can do. Conservation of energy tells us that the work is equal to the chemical potential energy of the gasoline. On average, a gallon of gasoline contains 132*10^{6} Joules of energy. To get total work, we just multiply the total gasoline in gallons by 132*10^{6}.

Thus, our final equation for the distance a vehicle can travel is:

d = 264*10^{6} / [C_{d}Aρv^{2} + 2μ_{k}N]

Next we will need to use some basic calculus to maximize the range difference between the two vehicles.

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