Resources for Velocity and Acceleration as Derivatives
-
Questions
12
With Worked SolutionClick Here -
Video Tutorials
1
Click Here -
HSC Questions
5
With Worked SolutionClick Here
Velocity and Acceleration as Derivatives Theory
![Given the displacement function \(x=f(t)\) for \(t=\) time. Then the velocity function is \(v=f^{\prime}(x)\) and the acceleration function is \(a=f^{\prime \prime}(t)\).\\ Alternative notation: If the displacement is \(x\), then the velocity function is \(\dot{x}\) and the acceleration function is \(\ddot{x}\).\\ \textbf{Example}\\ A particle moves along a straight line so that its distance \(x\), in metres, from a fixed point \(O\) is given by. $$x = 2t + 2\cos 2t$$ where the time \(t\) is measured in seconds from \(t = 0\). \begin{itemize}[nosep] \item[\bf{i)}]Where is the particle initially? \item[\bf{ii)}]When does the particle first come to rest? \item[\bf{iii)}]Where does the particle next come to rest? \item[\bf{iv)}]What is the acceleration of the particle when \(t = \dfrac{\pi}{4}\). \end{itemize} \textbf{Example 1 solution}\\ \begin{multicols}{2} i) \\ $\begin{aligned}[t] \text{when } t = 0 \;\;\; x&= 0 + 2\cos 0\\&= 2\text{ metres to the right of the origin.} \end{aligned}$\\ ii) \\ $\begin{aligned}x &= 2t + 2\cos 2t\\\dot{x} &= 2 - 4\sin 2t\\\text{let } \dot{x} &= 0\\\therefore 2 - 4\sin 2t &= 0\\\sin 2t &= \frac{1}{2}\\\therefore 2t &= \frac{\pi}{6}\\t &= \frac{\pi}{12} \text{ seconds.}\end{aligned}$\\ \columnbreak iii) \\ $\begin{aligned}\text{In (ii) consider } \sin 2t &= \frac{1}{2}\\\therefore 2t = \frac{\pi}{6} &\text{ or } \frac{5\pi}{6}\\t = \frac{\pi}{12} &\text{ or } \frac{5\pi}{12}\\\therefore x &= 2\times\frac{5\pi}{12} + 2\cos\frac{5\pi}{6} \\ \quad &\text{(choosing the later time)}\\&= \frac{5\pi}{6} + 2\times\frac{-\sqrt{3}}{2}\\&= \frac{5\pi}{6} - \sqrt{3}\text{ metres}.\end{aligned}$\\ iv) \\ $\begin{aligned}\ddot{x} &= -8\cos 2t\\\text{when } t = \frac{\pi}{4} &\quad \ddot{x} = -8\cos\frac{\pi}{2}\\\therefore \ddot{x} &= 0 \text{ metres/sec}^2.\end{aligned}$\\ \end{multicols}](/media/ghqnujpt/23164.png)