Find the velocity $\mathbf{v}$, acceleration $\mathbf{a}$, and speed $v$ of a particle that is moving along:

(a) the plane curve $\mathbf{r}(t)=\left( \dfrac{1}{2}t^{2}+t\right) \mathbf{i}+t^{3}\mathbf{j}$ from $t=0$ to $t=2$.

(b) the space curve $\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j}+t^{3}\mathbf{k}$ from $t=0$ to $t=2.$

For each curve, graph the motion of the particle and the vectors $\mathbf{v}(1)$ and $\mathbf{a}(1)$.

Figure 25 $\mathbf{r}(t)=\left(\dfrac{1}{2}t^2+t\right)\mathbf{i}+ t^3 \mathbf{j}$, \\ $0\le t\le 2$

Solution (a) The velocity, acceleration, and speed are $$\begin{eqnarray*} \mathbf{v}(t)&\;=\;&\mathbf{r}^{\prime} (t)=\dfrac{d}{dt}\left( \dfrac{1}{2} t^{2}+t\right) \mathbf{i}+\dfrac{d}{dt}t^{3}\mathbf{j}=(t+1)\mathbf{i}+3t^{2} \mathbf{j} \\[6pt] \mathbf{a}(t)&\;=\;&\mathbf{r}^{\prime \prime} (t)=\dfrac{d}{dt}\mathbf{r^{\prime} } ( t)\;=\;\dfrac{d}{dt}(t+1)\mathbf{i}+\dfrac{d}{dt}3t^{2}\mathbf{j=i} +6t\mathbf{j} \\[6pt] v(t)&\;=\;&\left\Vert \mathbf{v}(t)\right\Vert\;=\;\sqrt{( t+1) ^{2}+( 3t^{2}) ^{2}}=\sqrt{9t^{4}+t^{2}+2t+1} \end{eqnarray*}$$

At $t=1$, the velocity is $\mathbf{v}(1)=\mathbf{r}^{\prime} (1)=2\mathbf{i}+3\mathbf{j}$ and the acceleration is $\mathbf{a}(1)=\mathbf{r}^{\prime \prime}(1)=\mathbf{i}+6\mathbf{j}$. Figure 25 illustrates the graph of $\mathbf{r}=\mathbf{r}( t)$, $\mathbf{v}(1)$, and $\mathbf{a}(1)$.

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(b) The velocity, acceleration, and speed of the particle are $$\begin{eqnarray*} \mathbf{v}(t)&\;=\;&\mathbf{r}^{\prime} (t)=\dfrac{d}{dt}t\mathbf{i}+\dfrac{d}{dt} t^{2}\mathbf{j}+\dfrac{d}{dt}t^{3}\mathbf{k}=\mathbf{i}+2t\mathbf{j}+3t^{2} \mathbf{k} \\[12pt] \mathbf{a}(t)&\;=\;&\mathbf{r}^{\prime \prime} (t)=\dfrac{d}{dt}\mathbf{r^{\prime} } ( t)\;=\;\dfrac{d}{dt}\mathbf{i}+\dfrac{d}{dt}( 2t) \mathbf{j}+\dfrac{d}{dt}( 3t^{2}) \mathbf{k}=2\mathbf{j}+6t \mathbf{k} \\[12pt] v(t)&\;=\;&\left\Vert \mathbf{v}(t)\right\Vert\;=\;\sqrt{1^{2}+( 2t) ^{2}+( 3t^{2}) ^{2}}=\sqrt{1+4t^{2}+9t^{4}} \end{eqnarray*}$$

Figure 26 $\mathbf{r}(t)= t \mathbf{i} + t^2 {\bf j} + t^3 \mathbf{k}$, $0\le t \le 2$

At $t=1$, the velocity is $\mathbf{v}(1)=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$ and the acceleration is $\mathbf{a}(1)=2\mathbf{j}+6\mathbf{k}$. See Figure 26.