A fairly recent paper resolves a large discrepancy in the internal energy utilized to fire a cannon as calculated by two inertial observers. Earth and its small reaction velocity must be considered in the system so that the change in kinetic energy is calculated correctly. This paper uses a car in a similar scenario, but considers the work done by forces acting over distances. An analysis of the system must include all energy interactions, including the work done on the car and especially the (negative) work done on Earth in a moving reference frame. This shows the importance of considering the force on Earth and the distance Earth travels. For calculation of work in inertial reference frames, the center of mass perspective is shown to be useful. We also consider the energy requirements to efficiently accelerate a mass among interacting masses.

1.
Igal
Galili
and
Dov
Kaplan
, “
Extending the application of the relativity principle: Some pedagogical advantages
,”
Am. J. Phys.
65
(
4
),
328
335
(
April
1997
).
2.
Download the online appendix at TPT Online under the “Supplemental” tab, https://doi.org/10.1119/1.5011835 .
3.
See Ref. 1, p.
330
.
4.
Brandon J.
Tefft
and
James A.
Tefft
, “
Galilean relativity and the work-kinetic energy theorem
,”
Phys. Teach.
45
,
218
220
(
April
2007
).
5.
The same type of increase in work has been observed in other papers, such as the following paper by the author whereby a mass that undergoes a constant acceleration requires three times the amount of work to double its velocity because the force must act through a larger distance:
Richard
Kaufman
, “
Thrust and propulsive efficiency from an instructive viewpoint
,”
Phys. Educ.
45
(
5
),
494
499
(
2013
).
6.
The mechanism is used to do work on the car instead of using the car’s engine since an engine cannot do any work to accelerate the car by the drive wheel. Although acceleration is not possible without the external force of friction at the drive wheel, the force does no work since it does not act through a distance. This has been discussed in papers such as the following:
Claude M.
Penchina
, “
Pseudowork-energy principle
,”
Am. J. Phys.
46
,
295
296
(
March
1978
);
Bruce Arne
Sherwood
, “
Pseudowork and real work
,”
Am. J. Phys.
51
,
597
602
(
July
1983
);
Arnold B.
Arons
, “
Developing the energy concepts in introductory physics
,”
Phys. Teach.
27
,
506
517
(
July
1989
);
Arvind
Kumar
, “
Pitfalls in elementary physics – 3. Work and energy
,”
Resonance
3
(
12
),
69
77
(
1998
);
Richard
Kaufman
, “
Pseudowork and real work for a car
,”
Int. J. Mech. Eng. Educ.
41
(
1
),
8
13
(
2013
).
7.
Although the center of mass remains close to Earth in the car and cannon examples, it need not in other examples. Consider two equal masses propelled by a spring between them. The center of mass does not remain close to either mass, although the center of mass for the system of masses never changes for the interaction.
8.
A. John.
Mallinckrodt
and
Harvey. S.
Leff
, “
All about work
,”
Am. J. Phys.
60
,
356
365
(
April
1992
).
9.
Ibid., p.
359
.
10.
Ibid., p.
361
.
11.
Even when both objects undergo constant acceleration (via a constant force) for a period of time, the work done at different intervals of time can change as the distance traveled by different masses changes with velocity.
12.
For example, see the following paper:
Richard
Kaufman
, “
Thrust and propulsive efficiency from an instructive viewpoint
,”
Phys. Educ.
45
(
5
),
494
499
(
2010
).

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