# Modeling a car slowing down from speed

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1. Nov 15, 2014

### Belacan

Hello!

I'm trying to model a car slowing down from speed with the forces of drag and friction and gravity from the road slope..

The model, which models the resistive forces is as follows..

[ Ffriction + Fdrag + Fslope ] / m = -a (deceleration)

where:

Friction force
Ffriction = A where A = mg x Rd, m is mass, g is gravity Rd is the friction coefficient

Drag force
Fdrag = B v2
where B = 0.5 x ρ x A x Cd, ρ is density, A is frontal area, Cd is drag coefficient

Force due to slope
Fslope
= C sin(θs) where C = mg, θs is the angle of the road at displacement s

The displacement data is given in a discrete form.. for example,

θ0 = 2.2°
θ1 = 4.1°
θ2 = 3.2°
...
θn = x

My ultimate goal is to have an equation that gives me displacement as a function of time.. given the parameters A, B, the initial velocity vi and the road angle information..

I've had success in modeling the car without accounting for the slope by solving the following differential equation (from the model..)

-a = A + B v2

-s''(t) = A + B s'(t)2

Simply solving for s(t) (with wolfram alhpa..) I get an expression for displacement as a function of time and the coefficients A & B as well as constants on integration C1 & C2..

My question is.. how can I incorporate the road angle information?

It would be easier the road angle information could be provided as a function of time, θ(t) instead of θ(s).. but both should be workable..

Thanks!

2. Nov 15, 2014

### Stephen Tashi

Is the road actually a set of straight line segments, each with a constant slope? If not, I suggest fitting a smooth function to the slope data that approximates the actual shape of the road.

3. Nov 15, 2014

### Belacan

Hi Stephan, good question.

Indeed the road is not a set of straight line segments so it would make sense to do as you suggested.. I imagine this may be a necessary part of the solution as it would make sense the solution would require you to integrate the equation of the road somehow..