Find the velocity and acceleration of r=(6t3+2t)i + (t2+3t3)j + 12t4 at t=12 sec. Then find the difference of velocity at t=10 and t=12. 2. Find (P^Q) r use truth table.

Kwanza, rahisiisha vekta ya nafasi iliyopewa:

$$\vec{r}(t) = (6t^3 + 2t)\hat{i} + (t^2 + 3t^2)\hat{j} + 12t^4\hat{k}$$ $$\vec{r}(t) = (6t^3 + 2t)\hat{i} + (4t^2)\hat{j} + 12t^4\hat{k}$$

Kasi ni derivative ya kwanza ya nafasi kwa heshima ya muda ($t$):

$$\vec{v}(t) = \frac{d\vec{r}}{dt}$$ $$\vec{v}(t) = \frac{d}{dt}(6t^3 + 2t)\hat{i} + \frac{d}{dt}(4t^2)\hat{j} + \frac{d}{dt}(12t^4)\hat{k}$$ $$\vec{v}(t) = (18t^2 + 2)\hat{i} + (8t)\hat{j} + (48t^3)\hat{k}$$

Kuongeza kasi ni derivative ya kwanza ya kasi kwa heshima ya muda ($t$):

$$\vec{a}(t) = \frac{d\vec{v}}{dt}$$ $$\vec{a}(t) = \frac{d}{dt}(18t^2 + 2)\hat{i} + \frac{d}{dt}(8t)\hat{j} + \frac{d}{dt}(48t^3)\hat{k}$$ $$\vec{a}(t) = (36t)\hat{i} + (8)\hat{j} + (144t^2)\hat{k}$$

Badilisha $t = 12$ katika $\vec{v}(t)$:

$$\vec{v}(12) = (18(12)^2 + 2)\hat{i} + (8(12))\hat{j} + (48(12)^3)\hat{k}$$ $$\vec{v}(12) = (18(144) + 2)\hat{i} + (96)\hat{j} + (48(1728))\hat{k}$$ $$\vec{v}(12) = (2592 + 2)\hat{i} + 96\hat{j} + 82944\hat{k}$$ $$\vec{v}(12) = \boxed{2594\hat{i} + 96\hat{j} + 82944\hat{k} \text{ m/s}}$$

Badilisha $t = 12$ katika $\vec{a}(t)$:

$$\vec{a}(12) = (36(12))\hat{i} + (8)\hat{j} + (144(12)^2)\hat{k}$$ $$\vec{a}(12) = (432)\hat{i} + 8\hat{j} + (144(144))\hat{k}$$ $$\vec{a}(12) = \boxed{432\hat{i} + 8\hat{j} + 20736\hat{k} \text{ m/s}^2}$$

Badilisha $t = 10$ katika $\vec{v}(t)$:

$$\vec{v}(10) = (18(10)^2 + 2)\hat{i} + (8(10))\hat{j} + (48(10)^3)\hat{k}$$ $$\vec{v}(10) = (18(100) + 2)\hat{i} + (80)\hat{j} + (48(1000))\hat{k}$$ $$\vec{v}(10) = (1800 + 2)\hat{i} + 80\hat{j} + 48000\hat{k}$$ $$\vec{v}(10) = 1802\hat{i} + 80\hat{j} + 48000\hat{k} \text{ m/s}$$

Tofauti = $\vec{v}(12) - \vec{v}(10)$

$$\text{Tofauti} = (2594\hat{i} + 96\hat{j} + 82944\hat{k}) - (1802\hat{i} + 80\hat{j} + 48000\hat{k})$$ $$\text{Tofauti} = (2594 - 1802)\hat{i} + (96 - 80)\hat{j} + (82944 - 48000)\hat{k}$$ $$\text{Tofauti} = \boxed{792\hat{i} + 16\hat{j} + 34944\hat{k} \text{ m/s}}$$