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 $7m$.

(a) Find the distance traveled by the particle.

(b) Find the displacement (magnitude and direction).

(c) What would be these quantities if the particle completes the circle?

Answer 1

Step 1: Given data

Radius, $r=7m$.

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

Step 2: Find the distance traveled by the particle

Distance:

1. It is the length of the complete path traveled by a body.
2. It is a scalar quantity as it depends upon the magnitude and not the direction.
3. It can have positive values.

Displacement:

1. It is the shortest path between the initial and final positions.
2. It is a vector quantity as it has a direction and magnitude.
3. It can be positive, negative, and even zero.

(a) As the particle travels the path $A–C–B$, the distance covered by the particle will be equal to the circumference of the semi-circle.

Let the distance traveled be $d$.

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

$=7\pi m$

Step 3: Find the displacement of the particle

(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, $D=2r$

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

Step 4: Find the distance and displacement of the particle if the same path is a complete circle.

(c) 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 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.

Thus:

(a) The distance traveled by the particle on a semi-circular path is $\mathbf{7}\mathit{\pi }\mathbf{}\mathit{m}$.

(b) The displacement on a semi-circular path is $\mathbf{14}\mathbf{}\mathit{m}$ from A to B.

(c) The distance is $\mathbf{14}\mathit{\pi }\mathbf{}\mathit{m}$ and displacement is zero if the particle completes the circle.