Vectors, velocity and reference frames

A scalar has only a size (distance, speed, time). A vector has a size and a direction (displacement, velocity, acceleration, force). Choose a positive direction before you start, and in one dimension let the sign carry the direction.

Displacement is final position minus initial position, $\Delta x = x_f - x_i$; distance is the length of the path. So average velocity is $\Delta x/\Delta t$, while average speed is distance divided by time. The two have the same size only for one-way motion along a straight line.

In two dimensions, split each vector into perpendicular components. A vector of size A at angle θ from the x-axis has $A_x = A\cos\theta$ and $A_y = A\sin\theta$. Add the components separately, then rebuild the result with $A = \sqrt{A_x^2 + A_y^2}$ and $\tan\theta = A_y/A_x$.

Every velocity is measured relative to something. If A moves at $\vec v_{AB}$ relative to B, and B moves at $\vec v_{BC}$ relative to C, then A moves at $\vec v_{AC} = \vec v_{AB} + \vec v_{BC}$ relative to C. Add the velocities as vectors: along one line, add them with their signs; at right angles, use the Pythagorean theorem.

When you explain a result, name the frame of each velocity: relative to the walkway, relative to the ground, relative to the air. Most relative-motion mistakes come from mixing two frames in one equation.