SQA Higher Physics

## Distance, displacement, speed and velocity A **scalar** has size only; a **vector** has size and direction. Distance and speed are scalars. Displacement and velocity are vectors, and so is acceleration. The distance travelled is the total length of the path, however it twists. The **displacement** is the straight-line distance from the start to the finish, in a stated direction. Walk 30 m east and then 10 m west and the distance is 40 m but the displacement is 20 m east. Speed and velocity are found in the same way, with the distance or displacement divided by the time taken: $$ d = vt \qquad\qquad s = vt $$ Here $d$ is the distance in metres (m), $s$ is the displacement in metres, $v$ is the speed or velocity in metres per second ($\mathrm{m\,s^{-1}}$) and $t$ is the time in seconds (s). Used over a whole trip these give an average. The speed at one instant is found from a very short time interval. **Acceleration** is the rate of change of velocity, in metres per second squared ($\mathrm{m\,s^{-2}}$). It is a vector, so an object that slows down in the direction it is moving has an acceleration in the opposite direction, and an object that changes direction at constant speed is also accelerating. ## Motion with constant acceleration For an object moving in a straight line with constant acceleration, five relationships link the quantities involved. $$ v = u + at $$ $$ s = ut + \tfrac{1}{2}at^{2} $$ $$ v^{2} = u^{2} + 2as $$ $$ s = \tfrac{1}{2}(u + v)t $$ $u$ is the…