1. **State the problem:** Find the average velocity of a particle moving along the x-axis with velocity function $v(t) = 2 - t^2$ over the time interval from $t=1$ to $t=3$. 2. **Formula for average velocity:** The average velocity over an interval $[a,b]$ is given by $$\text{Average velocity} = \frac{1}{b-a} \int_a^b v(t) \, dt$$ 3. **Apply the formula:** Here, $a=1$ and $b=3$, so $$\text{Average velocity} = \frac{1}{3-1} \int_1^3 (2 - t^2) \, dt = \frac{1}{2} \int_1^3 (2 - t^2) \, dt$$ 4. **Calculate the integral:** $$\int_1^3 (2 - t^2) \, dt = \int_1^3 2 \, dt - \int_1^3 t^2 \, dt$$ Calculate each separately: $$\int_1^3 2 \, dt = 2t \Big|_1^3 = 2(3) - 2(1) = 6 - 2 = 4$$ $$\int_1^3 t^2 \, dt = \frac{t^3}{3} \Big|_1^3 = \frac{3^3}{3} - \frac{1^3}{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$$ So, $$\int_1^3 (2 - t^2) \, dt = 4 - \frac{26}{3} = \frac{12}{3} - \frac{26}{3} = -\frac{14}{3}$$ 5. **Calculate average velocity:** $$\text{Average velocity} = \frac{1}{2} \times \left(-\frac{14}{3}\right) = -\frac{14}{6} = -\frac{7}{3}$$ 6. **Interpretation:** The average velocity from $t=1$ to $t=3$ is $-\frac{7}{3}$. **Final answer:** $-\frac{7}{3}$ (Option C)