Equation of motion in two dimensions

For motion at constant velocity, the vector equation of motion expresses the position at every instant as the initial position plus accumulated displacement:

$\vec{S}(t) = \vec{S}_A + \vec{v}\,(t - t_A)$

Projecting onto the two axes, this single vector equation splits into two scalar parametric equations, one for each coordinate:

$\begin{cases} S_x(t) = S_{A,x} + v_x\,(t - t_A) \\ S_y(t) = S_{A,y} + v_y\,(t - t_A) \end{cases}$

The common parameter is time $t$. Eliminating it between the two equations gives the trajectory equation in the plane $(S_x, S_y)$, i.e. the direct relation between the coordinates, independent of time. For uniform rectilinear motion the trajectory is a straight line: the body traces out aligned points, at constant velocity in both magnitude and direction.

Try it — interactive simulation

The ball chases a sequence of targets: drag the boxes to move them. The green arrow is the velocity, always tangent to the path; the blue arrow is the acceleration. The side graphs split the acceleration into its tangential component (which changes the speed) and its normal component (which changes the direction), and show the curvature of the path.

Drag the coloured targets: the ball chases them. Green arrow = velocity (always tangent to the path), blue = acceleration. The graphs show speed, tangential (red) and normal (blue) acceleration, and curvature.

Collegamenti

Argomenti: Vettori Concetti: Velocità media Metodi: Scomposizione in componenti cartesiane

Esercizi collegati: Il cane nel parco · Ranking velocità di quattro auto · Distanza dall’area del grafico v-t