Question

A car is moving according to the following distance D function over the range from \(\mathrm{t}=0\) to

$$ \begin{array}{l} t=1=\pi / \omega \\ D(t)=L\sin (\omega t) \end{array} $$

Where \(L\) is the amplitude of the movement or displacement or linear distance, \(\omega\) is the angular frequency of the movement and \(t\) is the time. Confining this example to linear movement, then \(\mathrm{D}\) is the distance and \(0<\mathrm{D}<\mathrm{L}=1\) for \(0<\mathrm{t}<\pi / \omega\).

a) Derive expressions for the distance \(\mathrm{D},\) velocity \(\mathrm{V}\) and acceleration \(\mathrm{A}\) as a function of

time t.

b) Plot the variation of the distance \(D\), velocity \(V\) and acceleration \(A\) as a function of

time t for a linear displacement from \(\mathrm{t}=0\) to \(\mathrm{t}=1\).

c) Find the cost function \(\mathrm{C}(\mathrm{D}, \omega)\) using the results of the accelerometer readings, which are given in the following table: