The law of motion of a point is s = sin t, where s is

The law of motion of a point is $s = \sin t$, where $s$ is the distance in centimetres and $t$ is the time in seconds. Find the average velocity of the point over the time interval from $t = 0$ to $t = \dfrac{\pi}{2}$.

Step 1: Identify the time endpoints

\[ t_1 = 0, \qquad t_2 = \frac{\pi}{2}, \qquad \Delta t = t_2 - t_1 = \frac{\pi}{2}\ \text{s}. \]

Step 2: Compute the positions

\[ s_1 = \sin t_1 = \sin 0 = 0\ \text{cm}, \qquad s_2 = \sin t_2 = \sin\frac{\pi}{2} = 1\ \text{cm}. \]

The displacement over the interval is

\[ \Delta s = s_2 - s_1 = 1 - 0 = 1\ \text{cm}. \]

Step 3: Compute the average velocity

The average velocity is defined as

\[ v_{\text{avg}} = \frac{\Delta s}{\Delta t} = \frac{s_2 - s_1}{t_2 - t_1}. \]

Substituting: $v_{\text{avg}} = \frac{1}{\dfrac{\pi}{2}} = \frac{2}{\pi}\ \text{cm/s}$.

Step 4: Verification

\[\Delta s = \sin\!\left(\tfrac{\pi}{2}\right) - \sin 0 = 1 - 0 = 1\ \text{cm}\]

\[\frac{\Delta s}{\Delta t} = \frac{1}{\pi/2} = \frac{2}{\pi} \approx 0.6366\ \text{cm/s} \;\;\checkmark\]