Question

A particle traveled along a semi-circular path $A–C–B$ as shown in the figure below. The radius of the semi-circular path is $7m$.

Find the speed and velocity in each case if the time is taken

(a) to travel from A to B is $11s$.

(b) to complete the circle is $22s$

Answer 1

Step 1: Given data

The radius of the semicircular path,$r=7m$

Time taken to travel from A to B, $t_1=11s$

Time taken to complete the circle, $t_2=22s$

The semi-circular path traveled by the particle is given below:

(Note: The value of $\mathrm{\pi }$ is taken as $\frac{22}{7}$ for this solution).

Step 2: Find the speed of the particle from A to B

(a) As the particle travels from A to B in $11s$, the distance covered by the particle will be equal to the circumference of the semi-circle.

Let the distance traveled be $d$.

Therefore, $d=\mathrm{\pi r}$

$=7\pi m$

As we know, $speed=\frac{\mathrm{distance}}{\mathrm{time}}$.

$\Rightarrow speed=\frac{d}{t_1}$

$$\begin{gathered} =\frac{7\mathrm{\pi }}{11} \\ =\frac{7\times {\displaystyle \frac{22}{7}}}{11} \\ =2m{s}^{-1} \end{gathered}$$

Step 3: Find the velocity of the particle from A to B

The initial position of the particle is A and the final position is B.

Hence, the displacement will be equal to the diameter of the semicircle and its direction will be from A to B.

Let the displacement of the particle be $D$.

Therefore, $D=2r$

$$\begin{gathered} =2\times 7 \\ =14m \end{gathered}$$

As we know, $velocity=\frac{\mathrm{dis}placement}{\mathrm{time}}$.

$\Rightarrow velocity=\frac{D}{t_1}=\frac{14}{11}=1.27m{s}^{-1}$

Step 4: Find the speed of the particle when it covers the complete circle

(b) The total distance will be equal to the circumference of the complete circle in this case.

Therefore, $d=2\mathrm{\pi r}$

$$\begin{gathered} =2\times 7\pi \\ =14\pi m \end{gathered}$$

As we know, $speed=\frac{\mathrm{distance}}{\mathrm{time}}$.

$\Rightarrow speed=\frac{d}{t_1}$

$$\begin{gathered} =\frac{14\mathrm{\pi }}{11} \\ =\frac{14\times {\displaystyle \frac{22}{7}}}{11} \\ =4m{s}^{-1} \end{gathered}$$

Step 5: Find the velocity of the particle when it covers the complete circle

As the initial and final position of the particle will be the same in the case of a complete circular path, the displacement will be zero.

As we know, $velocity=\frac{\mathrm{dis}placement}{\mathrm{time}}$.

$\Rightarrow velocity=\frac{D}{t_1}=0m{s}^{-1}$

Thus:

(a) The speed of the particle is $2m{s}^{-1}$ and velocity is $1.27m{s}^{-1}$.

(a) The speed of the particle is $4m{s}^{-1}$ and velocity is $0m{s}^{-1}$.