The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles

Thanassis Dialynas

doi:doi:10.11648/j.ajpa.20210902.12

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The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles

Thanassis Dialynas

Published in American Journal of Physics and Applications (Volume 9, Issue 2)

Received: 2 February 2021 Accepted: 24 February 2021 Published: 26 April 2021

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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_r(t)=(rpm)(t) – (rpm)₀ 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_{c, 6}, b₆, σ₆, λ₆ which can be used, post – hoc, to examine together with δ_r(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^th gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.

Published in	American Journal of Physics and Applications (Volume 9, Issue 2)
DOI	10.11648/j.ajpa.20210902.12
Page(s)	34-41
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Motion of a Vehicle, Newton’s Second Law, Ordinary Differential Eqs

References

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[2]	J. K. Hale, Ordinary Differential Eqs - Wiley Interscience London 1971.
[3]	S. Trahanas, Ordinary Differential Eqs - Cretan University Press. Heraklion - Crete -Greece 1989.
[4]	Lawrence Perko, Differential Eqs and Dynamical Systems, Springer 2000.
[5]	The physics of racing: Weight transfer 2016, Allen Bery.
[6]	Motion & Forces : Newton’s Second Law of motion / Perkins.
[7]	Newton’s Laws of Motion & Car Physics, Carly Hallman.
[8]	Car Basics – Sir Isaac Newton’s Laws of Motion Quinton Kenrick.
[9]	The science Behind safe driving.
[10]	Newton’s Second Law – University Physics (Volume 1)
[11]	Car Physics and Newton’s Laws of Motion – Effects.
[12]	Seat Belts & Newton’s Second Law of Motion – Chris Deziel.
[13]	M. G. Calkin, “The motion of an accelerating automobile,” Am. J. Phys. 58, 573–575 (June 1990).
[14]	R. G. Carson, “Everyday physics: Auto test report,” Phys. Teach. 21, 369–375 (Sept. 1983).
[15]	J. Güémez and M. Fiolhais, “Forces on wheels and fuel consumption in cars,” Eur. J. Phys. 34, 1005–1013 (2013).
[16]	J-P Meyn, “The kinematic advantage of electric cars,” Eur. J. Phys. 36, 065037 (2015).
[17]	R. Cross, “Rolling to a stop down an inclined plane,” Eur. J. Phys. 36, 065047 (2015).
[18]	B. A. Sherwood, “Pseudowork and real work,” Am. J. Phys. 51, 597–602 (July 1983).
[19]	D. Tabor, “The rolling and skidding of automobile tyres,” Phys. Educ. 29, 301–306 (1994).
[20]	C. M Haaland, “Minimum engine size for optimum automobile acceleration,” Am. J. Phys. 60, 415–422 (May 1992).

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APA Style

Thanassis Dialynas. (2021). The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. American Journal of Physics and Applications, 9(2), 34-41. https://doi.org/10.11648/j.ajpa.20210902.12

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ACS Style

Thanassis Dialynas. The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. Am. J. Phys. Appl. 2021, 9(2), 34-41. doi: 10.11648/j.ajpa.20210902.12

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AMA Style

Thanassis Dialynas. The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. Am J Phys Appl. 2021;9(2):34-41. doi: 10.11648/j.ajpa.20210902.12

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@article{10.11648/j.ajpa.20210902.12,
  author = {Thanassis Dialynas},
  title = {The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles},
  journal = {American Journal of Physics and Applications},
  volume = {9},
  number = {2},
  pages = {34-41},
  doi = {10.11648/j.ajpa.20210902.12},
  url = {https://doi.org/10.11648/j.ajpa.20210902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20210902.12},
  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 δr(t)=(rpm)(t) – (rpm)0 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 Fc, 6, b6, σ6, λ6 which can be used, post – hoc, to examine together with δr(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 6th gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.},
 year = {2021}
}

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TY  - JOUR
T1  - The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles
AU  - Thanassis Dialynas
Y1  - 2021/04/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajpa.20210902.12
DO  - 10.11648/j.ajpa.20210902.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 34
EP  - 41
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20210902.12
AB  - 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 δr(t)=(rpm)(t) – (rpm)0 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 Fc, 6, b6, σ6, λ6 which can be used, post – hoc, to examine together with δr(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 6th gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.
VL  - 9
IS  - 2
ER  -

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Author Information

Thanassis Dialynas

Department of Physics, University of Crete, Heraklion, Greece

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Plain Text BibTeX RIS

APA Style

Thanassis Dialynas. (2021). The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. American Journal of Physics and Applications, 9(2), 34-41. https://doi.org/10.11648/j.ajpa.20210902.12

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ACS Style

Thanassis Dialynas. The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. Am. J. Phys. Appl. 2021, 9(2), 34-41. doi: 10.11648/j.ajpa.20210902.12

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AMA Style

Thanassis Dialynas. The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles. Am J Phys Appl. 2021;9(2):34-41. doi: 10.11648/j.ajpa.20210902.12

Copy | Download

@article{10.11648/j.ajpa.20210902.12,
  author = {Thanassis Dialynas},
  title = {The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles},
  journal = {American Journal of Physics and Applications},
  volume = {9},
  number = {2},
  pages = {34-41},
  doi = {10.11648/j.ajpa.20210902.12},
  url = {https://doi.org/10.11648/j.ajpa.20210902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20210902.12},
  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 δr(t)=(rpm)(t) – (rpm)0 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 Fc, 6, b6, σ6, λ6 which can be used, post – hoc, to examine together with δr(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 6th gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - The Wheel – Vehicle Considered as a Material Point: An Approximate Theory of Automobiles
AU  - Thanassis Dialynas
Y1  - 2021/04/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajpa.20210902.12
DO  - 10.11648/j.ajpa.20210902.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 34
EP  - 41
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20210902.12
AB  - 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 δr(t)=(rpm)(t) – (rpm)0 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 Fc, 6, b6, σ6, λ6 which can be used, post – hoc, to examine together with δr(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 6th gear, giving for power -45HP which is a very reasonable estimate in order of magnitude.
VL  - 9
IS  - 2
ER  -

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