Lat. Am. J. Phys. Educ. Vol. 16, No. 2, June, 2022 2309-1 http://www.lajpe.org ISSN 1870-9095 Kicking a cold soccer ball Carl E. Mungan 1 and Trevor C. Lipscombe 2 1 Physics Department, U.S. Naval Academy, Annapolis, Maryland, 21402, USA 2 Catholic University of America, Washington, DC, 20064, USA E-mail: mungan@usna.edu (Received 3 May 2022, accepted 26 June 2022) Abstract A soccer ball is conceptually split into halves so that it can be modeled as two identical blocks connected by a spring. A constant kicking force applied to one block simultaneously compresses the spring and imparts velocity to the center of mass of the system. If the half in contact with the foot detaches from it when it reaches a specified speed, the final center- of-mass velocity depends on the spring constant. That could explain why it is hard to kick a soccer ball as far on a cold day (when the spring is stiffer) as on a warm day. Keywords: Physics of sports, Newton’s second law, Mass-spring oscillator. Resumen Una pelota de fútbol se divide conceptualmente en dos mitades para que se pueda modelar como dos bloques idénticos conectados por un resorte. Una fuerza de patada constante aplicada a un bloque comprime simultáneamente el resorte e imparte velocidad al centro de masa del sistema. Si la mitad en contacto con el pie se separa cuando alcanza una velocidad específica, la velocidad final del centro de masa depende de la constante del resorte. Eso podría explicar por qué es difícil patear un balón de fútbol tan lejos en un día frío (cuando la primavera está más rígida) que en un día cálido. Palabras clave: Física de los deportes, Segunda ley de Newton, Oscilador masa-resorte. I. INTRODUCTION Football players report is it more difficult to kick a long field goal when the weather is cold. Data from the NFL corroborates this statement [1]. The longest kicks in history were either at Mile High Stadium in Denver, with an altitude of over 1.5 km above sea level, or else in warmer indoor or outdoor conditions. Mile High Stadium, where three of the five longest kicks took place, has reduced air density, thereby lowering the drag on the ball. But how can one understand the increased difficulty of kicking a ball in cold weather, when some players report it feels like kicking a rock? Sports balls have previously been analyzed as a single mass on a spring, with losses of mechanical energy represented by a coefficient of restitution [2] or by a viscous damper [3]. The present paper instead proposes a loss-free model of a kicked ball as two masses (representing the kicked hemisphere and the opposite hemisphere of the ball) connected by a spring (representing the elasticity of the ball). The ideas are thus within the grasp of an elementary physics student. The equations are solved theoretically and compare favorably with available soccer ball measurements. This approach may appeal to a physics of sports class, or as an example in a general introductory physics course when discussing the concept of mechanical oscillations. II. THEORETICAL MODEL Two blocks of mass m on a frictionless surface are connected by a massless spring of stiffness constant k. Initially the two blocks are at rest and the spring is relaxed. Starting at 0 t  a constant force F is applied to the left block 1 in the direction of the right block 2 in Fig. 1 until some later time f t t  when the left block has attained speed . What is the speed of the center of mass (cm) of the system at that final instant in time? m m k F x 1 x 2 X = 0 FIGURE 1. Two identical blocks of mass m are connected by a massless Hookean spring of stiffness constant k. A constant force F pushes the left block 1 toward the right block 2, but no other external horizontal forces act on the system. The displacements of the left block, right block, and center of mass of the system are respectively x 1 , x 2 , and X. Choose a coordinate axis-pointing positive rightward with the origin at the initial position of the cm. At some arbitrary instant in time t such that f 0 t t   , block 1 has displaced