For constant velocity, r(t)= r_0+t v. Use compatible time and distance units and a clearly stated time origin. For a later start at t=t_0, use r(t)= r_0+(t-t_0) v only after that start. A shared clock does not mean equal travel durations. Example: r_A(t)= 2t t, r_B(t)= 6-t t, with metres and seconds. Horizontal positions agree when 2t=6-t, giving t=2; vertical positions agree then too. They collide at (4,2). A collision requires both coordinates to agree at the same allowed time. Intersecting paths alone are insufficient. If the two coordinate equations give different times, there is no collision. The separation at a shared time is | r_B(t)- r_A(t)|. Find the difference vector first and then its magnitude.

For constant velocity, \(\mathbf r(t)=\mathbf r_0+t\mathbf v\). Use compatible time and distance units and a clearly stated time origin.

For a later start at \(t=t_0\), use \(\mathbf r(t)=\mathbf r_0+(t-t_0)\mathbf v\) only after that start. A shared clock does not mean equal travel durations.

Example: \(\mathbf r_A(t)=\begin{pmatrix}2t\\t\end{pmatrix}\), \(\mathbf r_B(t)=\begin{pmatrix}6-t\\t\end{pmatrix}\), with metres and seconds. Horizontal positions agree when \(2t=6-t\), giving \(t=2\); vertical positions agree then too. They collide at \((4,2)\).

A collision requires both coordinates to agree at the same allowed time. Intersecting paths alone are insufficient. If the two coordinate equations give different times, there is no collision.

The separation at a shared time is \(|\mathbf r_B(t)-\mathbf r_A(t)|\). Find the difference vector first and then its magnitude.

Positions and collision times

Positions and collision times