Modeling a car slowing down from speed

[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..

[ F_friction + F_drag + F_slope ] / m = -a (deceleration)

where:

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

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

Force due to slope
F_slope = C sin(θ_s) where C = mg, θ_s 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°
...
θ_n = 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 v_i 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!

 

[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.

 

[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..

Thanks for the reply!