Derivative

Suppose $A$, $B$, $C$ and $D$ are bodies, which are set in motion at the same initial time $t$. We know how the position $s$ or speed $v$ of these bodies changes with time: $$\begin{array}{rl}A:s=\frac{1}{2}{t}^2+10t+1,& C:v=9t+15,\\ B:s=\frac{1}{3}{t}^3+t^2+4,& D:v=\frac{5}{2}{t}^2+3.\end{array}$$ Position $s$ is given in meters, time $t$ in seconds and speed $v$ in meters per second. Determine which body moves with the greatest acceleration at the time $t=1\mathrm{s}$. $$$$ Hint: Instantaneous velocity can be expressed as the derivative of a position function $s(t)$ with respect to time: $v(t)=\frac{\mathrm{d}s}{\mathrm{d}t}$, and the instantaneous acceleration can be expressed as the derivative of a function $v(t)$ with respect to time: $a(t)=\frac{\mathrm{d}v}{\mathrm{d}t}$. Since we can determine the velocity using the derivative of the position function $s(t)$, we as well can determine the acceleration using the second derivative of $s(t)$: $a(t)=\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\cdot \frac{\mathrm{d}s}{\mathrm{d}t}=\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{t}^2}$.

The motion of two bodies is given by equations $$s_1=\frac{3}{2}{t}^2+3t+2\text{,}{s}_2=\frac{1}{3}{t}^3+\frac{t^2}{2}+1,$$ where the positions $s_1$ and $s_2$ are given in meters and the time $t$ in seconds. Determine at what time both bodies will move at the same speed. $$$$ Hint: Instantaneous velocity can be expressed as the derivative of a position function $s(t)$ with respect to time: $v(t)=\frac{\mathrm{d}s}{\mathrm{d}t}$.

Let’s have a coil of $0.06\mathrm{H}$ inductance. The current flowing through the coil is given by $$i=0.2\sin (100\pi t),$$ where time $t$ is measured in seconds and current $i$ is measured in amperes. Determine the voltage induced in the coil at time $t=2$ seconds. (Hint: Instantaneous voltage can be expressed as the derivative of current function with respect to time: $u(t)=-L\frac{\mathrm{d}i}{\mathrm{d}t}$. The negative sign indicates only that voltage induced opposes the change in current through the coil per unit time. It does not affect the magnitude of the voltage.)