AP Physics C: Mechanics

## Magnitude and direction A **scalar** is a quantity with a size and nothing else: mass, time, distance, speed, energy and temperature are examples. A **vector** has a size and a direction, and two vectors are equal only if both agree. Position, displacement, velocity, acceleration, force, momentum and torque are vectors. The size of a vector is its **magnitude**, and it is never negative. A vector is drawn as an arrow. The arrow points in the direction of the vector and its length is proportional to the magnitude, so a velocity arrow twice as long stands for twice the speed. Arrows can be moved about the page without changing the vector, provided the length and direction stay the same, which is what makes graphical addition possible. Distance and speed are scalars; the matching vectors are displacement and velocity. A runner who completes one lap of a track has covered a distance of 400 m and has a displacement of zero, because displacement compares only the start and the finish. The average speed over the lap is positive, and the average velocity is zero. In a one-dimensional problem a vector has only two possible directions, and these are told apart by sign. Once a positive direction has been chosen, a vector pointing the other way takes a negative sign, so a velocity of $-4\ \mathrm{m/s}$ means 4 m/s in the negative direction. The sign belongs to the component along the chosen axis and says nothing about whether the object is speeding up or slowing down. ## Unit…