Vectors, Reference Frames and Relative Motion

Scalars, vectors and components

A scalar has only a size (distance, speed, time); a vector has a size and a direction (displacement, velocity, acceleration, force). AP Physics 1 never asks you to add vectors by drawing carefully — it asks you to split them into components and add the components:

The angle in these formulas is always measured from the $x$-axis. If the problem gives an angle from the vertical, swap sine and cosine or, safer, redraw the angle first.

Reference frames

Every velocity is measured relative to something. A passenger walking at $1$ m/s toward the front of a train moving at $30$ m/s has velocity $31$ m/s relative to the ground and $1$ m/s relative to the train. Both are correct; the question is only which frame the question is asking about. The rule is a chain of subscripts:

Read it as "A relative to C equals A relative to B plus B relative to C". The inner subscripts cancel, exactly like units. In one dimension this is ordinary addition with signs; in two dimensions it is component addition.

The river-crossing problem

A boat that moves at $4$ m/s relative to still water heads straight across a river flowing at $3$ m/s. Relative to the shore its velocity is the vector sum: $4$ m/s across and $3$ m/s downstream, so its speed is $\sqrt{4^2+3^2}=5$ m/s and it drifts. Two facts matter for every version of this problem: the crossing time depends only on the across-stream component ($t=d/4$ here), and the downstream drift is that time multiplied by the current. Pointing the boat partly upstream cancels the drift but lengthens the crossing time, because the across-stream component is now $4\cos\theta \lt 4$.