Constant-Velocity Vector Motion (HL only)

For a particle moving with constant velocity \(\mathbf v\), position at time \(t\) is \(\mathbf r(t)=\mathbf r_0+t\mathbf v\), with speed \(|\mathbf v|\). Use compatible units for time and coordinates.

Two paths may intersect without a collision. For a collision, both position vectors must agree at the same time. For a path intersection, use independent parameters.

Example

\(\mathbf r_A(t)=\begin{pmatrix}0\\0\end{pmatrix}+t\begin{pmatrix}2\\1\end{pmatrix},\quad\mathbf r_B(t)=\begin{pmatrix}6\\0\end{pmatrix}+t\begin{pmatrix}-1\\1\end{pmatrix}\).

\(2t=6-t\) gives \(t=2\), and both second coordinates are \(2\). The particles collide at \((4,2)\). Their speeds are \(\sqrt5\) and \(\sqrt2\) distance units per time unit.