Deriving a differential equation for car motion

[cytochrome]

I'm looking at this scenario where a car is moving and then shifts into neutral. Knowing the initial velocity, how can I derive a differential equation?

I know the air drag and the frictional force... are there any other forces, like gravity, that should be included to make it realistic? I don't understand what a downward gravitational force would do for this problem

 

[HallsofIvy]

"Downward gravitational force" (weight) would affect the tire traction. But if you are assuming the tires are rotating freely and not dragging on the ground (no braking or acceleration), that should not be an issue. There will, of course, be friction in the wheel bearings.

 

[Chestermiller]

The dominant force is going to be the air drag. That should make things easy for you, if the air drag can be parameterized accurately.

Chet