Basketball
Trajectory of a basketball when shooting
Autor:
Título 4
Learning objectives :
This activity allows students to analyze the parabolic trajectory of a basketball during a shot. It develops skills in mathematical modeling and kinematic analysis.
Concepts covered
Parabolic trajectory; Projectile movement; Quadratic interpolation; Severity; Measurement accuracy
What students will do :
The student uses the FizziQ Kinematic Study module to analyze a basketball shooting video. After calibrating the scale and pointing the position of the ball frame by frame, the student obtains position data which he analyzes by applying mathematical interpolation to verify the parabolic nature of the trajectory and evaluate the precision of his pointing.
What is required :
Smartphone or tablet with the FizziQ application; 'Basketball' video accessible via FizziQ resources; FizziQ experience notebook
Scientific background :
Basketball shooting perfectly illustrates the laws of Newtonian mechanics applied to the movement of projectiles. Once released, the ball follows an essentially parabolic trajectory, influenced primarily by two forces: gravity (constant, directed downward) and air resistance (generally negligible for a basketball over this distance). The mathematical form of this trajectory is a parabola described by the parametric equations: x(t) = x₀ + v₀ₓt and y(t) = y₀ + v₀ᵧt - ½gt², where (x₀, y₀) is the initial position, (v₀ₓ, v₀ᵧ) the components of the initial velocity, g the acceleration of gravity (9.81 m/s²) and t the time. Quadratic interpolation of the curve y(x) gives an equation of the form y = ax² + bx + c, where the coefficient a is directly linked to g and to the initial horizontal speed by the relation a = -g/(2v₀ₓ²). The kinematic analysis with FizziQ makes it possible to experimentally verify this relationship and to indirectly estimate the initial speed of the shot. To make a basket, the player must intuitively solve a complex ballistics problem, taking into account the distance to the basket, the height of the latter (3.05 m), and his own height. For a shot at 6.75 m (three-point line), the optimal angle is approximately 45-50° and the necessary initial speed is approximately 7-8 m/s. Adjusting the interframe interval (176 ms) in the analysis helps optimize accuracy: intervals that are too short may capture positions that are too close, increasing the relative pointing error, while intervals that are too long may miss important details of the trajectory.