Accelerated motion and projectile motion

The equations of uniformly accelerated motion extend naturally to the two-dimensional case by replacing scalars with vectors:

Equations of accelerated motion in 2D

$\vec{S}_B - \vec{S}_A = \vec{v}_A\,(t_B - t_A) + \frac{1}{2}\,\vec{a}\,(t_B - t_A)^2$ $\vec{v}_B - \vec{v}_A = \vec{a}\,(t_B - t_A)$

In components, the equation of motion becomes a pair of independent equations:

$\begin{cases} S_x(t) = S_{A,x} + v_{A,x}\,(t - t_A) + \frac{1}{2}\,a_x\,(t-t_A)^2 \\ S_y(t) = S_{A,y} + v_{A,y}\,(t - t_A) + \frac{1}{2}\,a_y\,(t-t_A)^2 \end{cases}$

The trajectory of a 2D motion with constant acceleration is a parabola. The most important case is projectile motion (oblique launch), in which the acceleration is entirely vertical, directed downwards:

$\vec{a} = \begin{pmatrix} 0 \\ -g \end{pmatrix}$

Here lies the key to projectile motion: along $x$ there is no acceleration, so the horizontal motion is uniform; along $y$ only gravity acts, so the vertical motion is free fall. The two components are independent and combine into the curved trajectory.

Velocity and acceleration, distinct roles

The velocity vector $\vec{v}_A$ is always tangent to the trajectory at point A, while the acceleration vector $\vec{a}$ “curves” the trajectory, bending it towards itself. In projectile motion $\vec{g}$ always points downwards, even while the body is rising.

Projectile motion: the velocity $\vec{v}$ is tangent to the trajectory, gravity $\vec{g}$ always points downwards.

Try it — interactive simulation

Launch a projectile: set the angle and initial speed, then press play. The horizontal component is uniform, the vertical one accelerated: they stay independent and together give the parabola. Blue arrows: total velocity and its vx, vy components; green arrow: gravity. Range R and maximum height marked in red, axes graduated in metres.

Collegamenti

Argomenti: Vettori Concetti: Moto parabolico · Caduta libera Metodi: Scomposizione in componenti cartesiane

Esercizi collegati: Caduta libera in due dimensioni · Calciatore gittata e altezza massima · Esercizio svolto — Gittata di un proiettile