The sums of \(\mathbf{m_i\cdot v_i}\) and \(\mathbf{m_i\cdot v_i^2}\) are preserved, why not sum of \(\mathbf{m_i\cdot v_i^3}\): a pedagogical remark

Mcclure J.; Kosheleva O.; Kreinovich V.

Mathematical Structures and Modeling 2018. N. 1(45). PP. 49-51

UDC 531:372.853 DOI: 10.25513/2222-8772.2018.1.49-51

THE SUMS OF Mi • Vi AND Mi • Vf ARE PRESERVED, WHY NOT SUM OF MI • V?: A PEDAGOGICAL REMARK

John McClure

Ph.D. (Eng.), Professor, e-mail: mcclure@utep.edu Olga Kosheleva Ph.D. (Phys.-Math.), Associate Professor, e-mail: olgak@utep.edu

Vladik Kreinovich Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu

University of Texas at El Paso, El Paso, Texas 79968, USA

Abstract. Students studying physics sometimes ask a natural question: the momentum - sum of mi • Vi - is preserved, the energy - one half of the sum of mi • vf - is preserved, why not sum of mi • v?? In this paper, we give a simple answer to this question.

Keywords: momentum conservation law, energy conservation law, teaching physics.

1. Formulation of the problem

Students studying physics sometimes ask a natural question:

• the momentum - sum of mi • Vi - is preserved,

• the energy - one half of the sum of mi • v\ - is preserved, so

• why not sum of mi • v??

In this paper, we give a simple pedagogical answer to this question.

2. Our explanation

To answer the above question, let us consider a simple 2-body 1-D mechanical problem. We have two small (point-like) solid objects with masses mi = 2 and mi = 1 on a straight line, the second object is to the left of the first one. Originally:

• the first object is at rest (its velocity is vi = 0), while

• the second object moves towards the first object with velocity v2 = 1. What will happen when the second object hits the first one?

The situation is invariant with respect to rotations around the line connecting the two objects, so the resulting trajectories will also be invariant - and thus, both objects will continue to move along the same line. The only remaining question is: what will be the new velocities v'i and v'2?

We know that the momentum is preserved, and we know that kinetic energy is preserved. Thus, we can conclude that

mi • v'i + m2 • v'2 = mi • vi + m2 • v2

(1)

50 J. McClure, O. Kosheleva, V. Kreinovich. WhyJ2 mi • tf Is Not Preserved?

and

mi • (vi)2 + m2 • (^2)2 = mi • v\ + m2 • (2)

Substituting the known values m\ = 2, m2 = 1, v\ = 0, and v2 = 1 into these formulas, we conclude that

2^1 + v2 = 1 (1a)

and

From (1a), we conclude that

2K )2 + (v2 )2 = 1. (2a)

4 = 1 - 2v[. (3)

Substituting this expression into the formula (2a), we get

2(4)2 + (1 - 2v[)2 = 1.

Opening the parentheses, we get

2(v[)2 + 1 - 4v[ + 4(^)2 = 1.

Subtracting 1 from both sides and bringing similar terms together, we get

6(v[)2 - 4v[ = 0,

v[ • (6v[ - 4) = 0.

i.e., equivalently,

Thus, we have two options:

• either v'1 = 0,

12

• or 6v' - 4 = 0, so that v\ = -.

i . i 3

In the first case, when v\ = 0, from the formula (3), we conclude that v'2 = 1. This means that the the first object remains immobile, and the second object continue with the same speed - so that it somehow passed through the first object. For solid objects (not for ghosts), this is not possible.

Since the case v[ = 0 is not physically possible, we are left with the second 21

option v[ = -. In this case, the formula (3) implies that 13

41

v'2 = 1 - 2v\ = 1 — = —. 2 1 3 3

One can check that in this case, the equalities (1) and (2) - describing preservation of the sums of mi • Vi and mi • v\ - are satisfied, while the sum of mi • if are not preserved: indeed, in this case,

m1 • v\ + m2 • vl = 2 • 03 + 1 • 13 = 1,

while

mi • W)3 + «2 • (t>2)3 = 2 •(3) 3 + (-3)

Mathematical Structures and Modeling. 2018. N. 1(45)

51

= 2 • — -1 = 16 - 1 = 15 = 1.

37 27 27 27 27 ^ In this example, we use specific weights and velocities, to make computations simpler, but we could get the same conclusion by taking almost any combination of initial masses and velocities.

Conclusion: we cannot require that the sum of the terms mi • vf is preserved: this way, there would be no way to describe a simple collision.

Acknowledgments

This work was supported in part by the National Science Foundation grant HRD-1242122 (Cyber-ShARE Center of Excellence).

References

1. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics. Boston : Addison Wesley, 2005.

СУММЫ Mi ■ Vi И Mi ■ V2 СОХРАНЯЮТСЯ, ПОЧЕМУ НЕ СОХРАНЯЕТСЯ СУММА MI ■ Vf: ПЕДАГОГИЧЕСКОЕ ЗАМЕЧАНИЕ

Дж. МакКлюр

к.т.н., профессор, e-mail: mcclure@utep.edu О. Кошелева к.ф.-м.н., доцент, e-mail: olgak@utep.edu В. Крейнович

к.ф.-м.н., профессор, e-mail: vladik@utep.edu

Техасский университет в Эль Пасо, США

Аннотация. Студенты, изучающие физику, иногда задают естественный вопрос: импульс — сумма т^ ■ ^ — сохраняется, - энергия - половина суммы т^ ■ V* — сохраняется, почему не сохраняется сумма т^ ■ V3? В этой статье мы даём простой ответ на этот вопрос.

Ключевые слова: закон сохранения импульса, закон сохранения энергии, преподавание физики.

Дата поступления в редакцию: 27.11.2017

The sums of \(\mathbf{m_i\cdot v_i}\) and \(\mathbf{m_i\cdot v_i^2}\) are preserved, why not sum of \(\mathbf{m_i\cdot v_i^3}\): a pedagogical remark Текст научной статьи по специальности «Философия, этика, религиоведение»

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