Détermination de l'équation de la trajectoire à partir des équations horaires

To determine the trajectory equation of a circular motion from the time equations, we can use the parametric equations of circular motion. Let an object move in a circle of radius R around the origin O in a Cartesian coordinate system (x, y). The time equations of motion are given by: [ x(t) = R cos(omega t + θ)] [ y(t) = R sin(omega t + θ)] where: R is the radius of the circle, θ is the angular velocity, t is time, θ is the initial phase (the angle between the object's position at the initial time and the x-axis). To determine the trajectory equation, we eliminate the time \( t \) between the two time equations. To do this, we can square and add the two equations: [ x(t)^2 + y(t)^2 = R^2 ] Substituting the time equations into this expression, we obtain: [ R^2 \cos^2(\omega t + \theta) + R^2 \sin^2(\omega t + \theta) = R^2 ] Using the trigonometric identities \( \cos^2(\theta) + \sin^2(\theta) = 1 ), we obtain: [ R^2 = R^2 ] This is true, so the resulting equation is the same as that for the trajectory of circular motion: [ x^2 + y^2 = R^2 ] This equation represents a circumference of radius \( R ) centered at the origin of the coordinate system.