Launch of Toy Projectiles: mathematical modelling and investigative experimentation

Abstract

Studies on projectile launching at various educational levels are based upon simplified theoretical models. We abstract the projectile as a particle moving in two dimensions and not subjected to the resistive force of the air. In this work, we propose a simple experimental procedure involving low-cost toys, whose main objective is to implement investigative activities in the classroom, awakening a critical posture about the theoretical models studied. We carry out a detailed review of two existing academic models for the oblique launching of a particle to demonstrate the relationship between mathematical modelling and experimentation: the standard model with no air resistance and the model subject to a drag force proportional to the particle velocity.  We employed the usual strategy by solving the first model through vectorial kinematics while developing the equations of motion directly from Newton's laws and solving them exactly for the second. We obtained, particularly, analytical expressions for the range of the projectiles as a function of the parameters to both models. Beyond comparing and discussing the results of the mathematical models, a low-cost investigative experimental approach is proposed for real projectiles launching using toys. During this procedure, we challenged the students to raise spontaneous conceptions about the validity of the model with no air resistance they had been previously presented in the classroom. Subsequently, we exhibit the results from the model with air resistance to the students, not necessarily explaining the details of the mathematical development, making considerations about some of the improvements. Ultimately, pieces of evidence are displayed about the efficiency of investigative activities in the understanding of whatever a mathematical model is and its relationship with reality.