In $1.0\,\text{s}$, a particle goes from point $A$ to point $B$, moving in a semicircle of radius $1.0\,\text{m}$ (see Figure). The magnitude of the average velocity

Step 1: Identify the given parameters.

The radius of the semicircle $R$ and the time interval $\Delta t$ are given.

$$R = 1.0\,\text{m}$$ $$\Delta t = 1.0\,\text{s}$$

Step 2: State the formula for average velocity.

Average velocity is the ratio of displacement to the elapsed time.

$$\langle \vec{v}\rangle = \frac{\Delta \vec{r}}{\Delta t}$$

Step 3: Determine the displacement.

The displacement $\Delta r$ is the straight-line distance from point $A$ to point $B$, which corresponds to the diameter of the semicircle.

$$\Delta r = 2R$$ $$\Delta r = 2(1.0\,\text{m}) = 2.0\,\text{m}$$

Step 4: Calculate the magnitude of the average velocity.

Divide the magnitude of the displacement by the time interval.

$$|\langle \vec{v}\rangle| = \frac{\Delta r}{\Delta t}$$ $$|\langle \vec{v}\rangle| = \frac{2.0\,\text{m}}{1.0\,\text{s}}$$ $$|\langle \vec{v}\rangle| = 2.0\,\text{m/s}$$

Answer: The magnitude of the average velocity is $2.0\,\text{m/s}$.