The Physics Teacher ◆ Vol. 50, May 2012 305

Physics Challenge for Teachers and Students

Boris Korsunsky, Column EditorWeston High School, Weston, MA 02493 korsunbo@post.harvard.edu

w Softball in Space

In outer space, two small balls of equal unknown masses and charges +q and –q are held at rest a distance do apart. Then the balls are simultaneously launched with equal speeds vo in the opposite directions that are perpendicular to the line con-necting the balls. During the subsequent motion of the balls, their minimum speed is v. Find the masses of the balls.

Solution: The diagrams below show the initial and subse-quent states of the two moving charges that in general will execute symmetrical elliptical orbits around the common center of mass. You can see that when the velocity is at a minimum, (a) the separation is a maximum, which I call d, and (b) the velocity vectors are perpendicular to the sepa-ration line, which makes it easy to compute the angular momentum.

To find the particles’ identical masses m in terms of q, d0, v0, and minimum speed v:

Like the gravitational force (which we neglect here), the electrostatic force is a central conservative force, so both mechanical energy E and angular momentum L are con-served.

Initially:

(1a, 1b)

22 21 1

2 20 00 0

0 0 0 0 0 0

4and 2 2

qE mv mv ,d

L mv d / mv d / mv d .

= + −

= + =

πε

Later, at the minimum speed v, since L = mvd is con-served, we can express the maximum separation d as

0 0L v ddmv v

= =

from Eq. (1b), and the conserved mechanical energy is then:

2 2

2 2

0 0 0 0

, i.e. =4 4q q vE mv E mvd v d

= − −πε πε

.

(2)

To find the unknown mass m, just set the mechanical energies in Eqs. (1a) and (2) equal.

2 22 20

0 0 0 0 02

2 20

00 02

00 0

0 0 02

0 0 0 0

. 4 4

Rearranging 1 4

4

14

q q vmv mvd v d

q vm( v v ) ( )vdq ( v v )m( v v )( v v )d v

qm .d v ( v v )

− = −

→ − = −

−→ − + =

→ =+

πε πε

πε

πε

πε

(Contributed by Philip Blanco, Grossmont College, El Cajon, CA)

We would also like to recognize the following contribu-tors:

Hratch Barsoumian (Haigazian University, Beirut, Lebanon)

Phil Cahill (Lockheed Martin Corporation, North York-shire, United Kingdom)

Solution to February 2012 Challenge

v0

d0

v0

v

d

(i) Initial separation d0 and velocities v0

(ii) Minimum velocities v, maximum separation d

306 The Physics Teacher ◆ Vol. 50, May 2012

Jorge Manuel Caravaca Vidal, student (Universidad de Sevilla, Sevilla, Spain)

Jaime Carrillo Moreno, student (Escuela Politécnica Supe-rior, Universidad de Sevilla, Sevilla, Spain)

Juan Díaz Vergara (Instituto Hebreo, Santiago, Chile)F. Javier Doblas (Escuela Técnica Superior de Ingeniería,

Universidad de Sevilla, Spain)Don Easton (Lacombe, Alberta, Canada)Oscar Escucha García (student, Escuela Politécnica

Superior, Universidad de Sevilla, Sevilla, Spain) Josh Gates (Tatnall School, Greenville, DE)Fredrick P. Gram (Cuyahoga Community College,

Cleveland, OH) Fernando Ferreira (Universidade da Beira Interior, Co-

vilhã, Portugal) Norge Cruz Hernández (Universidad de Sevilla, Sevilla,

Spain) Gerald E. Hite (TAMUG, Galveston, TX)Art Hovey (retired, Milford, CT)Teclu Cezar Iacob (Escuela Politecnica Superior, Sevilla,

Spain)José Ignacio Íñiguez de la Torre (Universidad de Salaman-

ca, Salamanca, Spain)Stephen McAndrew (Macquarie University, Sydney,

Australia) Daniel Mixson (Naval Academy Preparatory School,

Newport, RI)

Carl E. Mungan (U. S. Naval Academy, Annapolis, MD)

Thomas Olsen (Society of Physics Students, AIP, Col-lege Park, MD)

Israel Pérez Luna, student (Escuela Politécnica Supe-rior, Universidad de Sevilla, Sevilla, Spain)

Michael Rapport (Anne Arundel Community College, Arnold, MD)

Pascal Renault (John Tyler Community College, Midlothian, VA)

Gregory Ruffa (University of Minnesota, Minneapolis, MN)

Daniel Schumayer (University of Otago, Dunedin, New Zealand)

Jason L. Smith (Richland Community College, Decatur, IL)

Cássio dos Santos Sousa, student (Instituto Tecnológi-co de Aeronáutica, São Paulo, Brazil)

Clint Sprott (University of Wisconsin – Madison, WI)William Whitney (Dame Preparatory School, Towson,

MD)

Many thanks to all contributors and we hope to hear from you in the future!