SQA Advanced Higher Physics

## Rates of change Kinematics describes motion without asking what causes it. At Advanced Higher the quantities are treated as functions of time, and the language of calculus is used because velocity and acceleration are rates of change. Differential notation writes a rate of change as a derivative. If a quantity $y$ depends on time $t$, then $\dfrac{dy}{dt}$ is how fast $y$ is changing at that instant. It is the gradient of the graph of $y$ against $t$, taken at a single point, which is why it can be found for a curve as well as for a straight line. For straight-line motion, with displacement $s$ measured from a fixed point: $$ v = \frac{ds}{dt} \qquad a = \frac{dv}{dt} = \frac{d^2 s}{dt^2} $$ Velocity measures how quickly displacement changes as time passes, and acceleration how quickly velocity changes, so acceleration is the second derivative of displacement. $v$ is in $\mathrm{m\,s^{-1}}$ and $a$ in $\mathrm{m\,s^{-2}}$. Because displacement, velocity and acceleration are vectors, in one dimension their sign carries the direction: a negative $v$ means motion in the negative direction, and a negative $a$ with a positive $v$ means the object is slowing down. ## Deriving the equations of motion The equations for constant acceleration follow from the definitions by integration, so they do not need to be learned as separate facts. They apply only when $a$ is constant. Start from $a = \dfrac{dv}{dt}$ with $a$ constant, and let the velocity be $u$ at $t = 0$ and $v$ at time…