AP Physics 1

## Scalars and vectors in one dimension A **scalar** quantity is described by a magnitude alone: a number and a unit. A **vector** quantity needs a magnitude and a direction. Distance, speed, time, mass, energy and power are scalars. Position, displacement, velocity, acceleration, force and momentum are vectors. A vector is drawn as an arrow. The arrow points in the direction of the quantity and its length is proportional to the magnitude, so a velocity arrow twice as long stands for twice the speed. In print a vector is written with an arrow over the symbol, as in $\vec{v}$, and the symbol without the arrow usually means the magnitude. In one dimension the direction of a vector is carried entirely by its sign. A coordinate axis is chosen, one direction along it is called positive, and every vector component along the axis is then a positive or negative number. A car moving at $20\ \mathrm{m/s}$ to the right and a car moving at $20\ \mathrm{m/s}$ to the left have the same speed and opposite velocities, $v_x = +20\ \mathrm{m/s}$ and $v_x = -20\ \mathrm{m/s}$. The choice of positive direction is free, but it must be kept for every quantity in a problem. If up is positive for velocity, then up is positive for acceleration, force and displacement as well. ### Adding vectors in one dimension Vectors along the same line are added by adding their signed components. Opposite directions have opposite signs, so they partly or wholly cancel. A boat that can move at $6\ \mathrm{m/s}$…