Abstract
The objective of the article is the mathematical description of the car motion in the most possible general form using Newton’s second law and the forces that act on it when they are known. In the first section, the forces that act on the vehicle are described and the normal (usual) conditions of driving are considered. Secondly the dynamical equation of motion baced on Newton’s second law is introduced which is in general a non-linear second order ordinary differential equation. Various cases are discussed such as going uphill, downhill, accelerating, decelerating etc. In Section 3, the energy consumption of the fuel is discussed and it is stated that it is consisted of two parts this of the “idle” worke and this of the sustainment of the motion. Besides it is shown that for a certain space “s” there is one unique speed that minimizes the consumption of fuel. In Section 4, the basic “defect” of the equation of motion which is the inclusion in the equation of the unknown driving force F(t) it is shown that it can be “circumvented” with energetic considerations leading to an equation having at the right – hand side the speed in the denominator and the excess revolutions per minute in the numerator. The resulting equation is such that a knowledge of δ<sub>r</sub>(t)=(rpm)(t) – (rpm)<sub>0</sub> can, by the numerical solution of the equation, lead to the function of speed and so a relation is established detween the velocity (u(t)) and the excess (rpm) which can be cheched as true or false by the aposteriori resister of the tachograph (u(t)) and rotation – counter (rpm(t)). Finally, in Section 5, we calculate, using the decelerating motion of a car in a flat road (when somebody leaves the throttle) all the kinematical and “energetical” constants that are introduced in the previous sections for sixth gear such as F<sub>c, 6</sub>, b<sub>6</sub>, σ<sub>6</sub>, λ<sub>6</sub> which can be used, post – hoc, to examine together with δ<sub>r</sub>(t) if the real velocityof a vehicle coincides with the prediction that a computer can make. Besides for a flat road, the power of a car can be estimated for instance when it has u=120 km/h at rpm=3000 and in the 6<sup>th</sup> gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.
Highlights
IntroductionIn the last three decades, there is a considerable effort of physicists for understanding the physics of automobiles and related topics [5-20]
The basic tool is Newton’s Second Law which if it is combined with other considerations like the theory of o.d.e.’s can probably go far, giving a theory of automobile considered as a “material point”
If we impose all the forces that are exert on the car it is a matter of elementary differential geometry of a certain trajectory, to see that all the forces not tangentional to the road are used to keep the car in the certain trajectory, letting the tangential forces, to determine the velocity of the "material point" [1–4]
Summary
In the last three decades, there is a considerable effort of physicists for understanding the physics of automobiles and related topics [5-20]. A motion of a car is a motion of certain trajectory (for instance a closed road) which may be flat, may have turns or may pass through hills (uphill or downhill). If we impose all the forces that are exert on the car (considered as a material point) it is a matter of elementary differential geometry of a certain trajectory, to see that all the forces not tangentional to the road are used to keep the car in the certain trajectory, letting the tangential forces, to determine the velocity (speed) of the "material point" [1–4]. Since u(t) ≥ 0 we have only forward motion i.e. reverse velocity is excluded from our considerations. The forces exerted on the car by the “environment” (as a reaction) are such to keep the vehicle on the trajectory of the road 3. For the last case of steep downhill motion we have:.
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