Use vectors in geometry, relative motion and force models

Apply constant-motion reasoning

Use relative velocity, constant-velocity positions, collision and closest approach with a common frame and allowed time interval

State a frame before combining velocities

All position and velocity components must refer to compatible axes, units and times. In a fixed frame, constant velocity gives the position rule below. Subtract B's velocity and position from A's to obtain A relative to B. Speed is a magnitude, not a signed component

If vA=(4,1) and vB=(1,−2) m/s,find the x-component of A relative to B.

With r₀=(−1,2) m and constant v=(3,−1) m/s,find x after 4 s.

A crossing path is not necessarily a collision

Collision of ideal point particles requires equal positions at the same allowed time, as shown below. Solve both coordinate equations with one common time. A solution in the past is excluded. Equal velocities with distinct starting positions never meet; equal positions and equal velocities remain coincident

A follows rA(t)=(t,0) and B follows rB(t)=(2,t−1),for t≥0. Do they collide?

Closest approach uses relative motion

For nonzero relative velocity, minimise squared separation. Completing the square gives an unrestricted minimising time. For future motion, use the larger of that time and zero, as shown below. If relative velocity is zero, separation is constant and the earliest minimising time is zero

For r₀=(6,2) m and w=(−2,0) m/s,find the future time of closest approach

For starting separation (6,2) m and relative velocity (−2,0) m/s over nonnegative time, find the squared minimum distance in square metres

Justify the relative-motion test, ungraded

Derive the relative position formula and closest-approach rule without assuming the two paths intersect. Include the zero-relative-velocity case and the future-time restriction. Explain why a zero minimum distance is required for a future point-particle collision

Model: subtract positions and complete a square

Subtract B's position from A's position to obtain the relative position. For nonzero relative velocity, complete the square as shown below. The square is minimised at the unrestricted time; if that time is negative, the minimum over future time occurs at zero. If relative velocity is zero, all distances are the same. Future collision requires the minimum to be zero, with the same time used in both coordinates

Extend the model only when its assumptions change

For variable velocity, replace the constant-velocity position rule below with the supplied vector function or a componentwise velocity integral. The same-time and allowed-domain collision conditions still apply. The companion lesson extends these conditions to nonlinear crossings