Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.
A) $3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

B) $2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
C) $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
D) $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$
E) $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$
F) $2 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$

G) $4 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

H) $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

Question

Let the velocity of a particle be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$ , and let its position when t = 0 be $\mathbf { r } ( 0 ) = \mathbf { j } + 2 \mathbf { k }$ . Find its position when t = 2.
A) $3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

B) $2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$ 
C) $3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$ 
D) $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ 
E) $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$ 
F) $2 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$

G) $4 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$

H) $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$

Quizplus · Accepted Answer

The position function is the integral of the velocity function. Integrating $\mathbf{v}(t) = \mathbf{i} + t\mathbf{j}$ gives $\mathbf{r}(t) = t\mathbf{i} + \frac{t^2}{2}\mathbf{j} + C$. Using $\mathbf{r}(0) = \mathbf{j} + 2\mathbf{k}$, we find $C = \mathbf{j} + 2\mathbf{k}$. Thus, $\mathbf{r}(t) = t\mathbf{i} + \left(\frac{t^2}{2} + 1\right)\mathbf{j} + 2\mathbf{k}$. At $t = 2$, $\mathbf{r}(2) = 2\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}$.

Let the Velocity of a Particle Be v(t)=i+tj\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }v(t)=i+tj

Let the Velocity of a Particle Be $\mathbf { v } ( t ) = \mathbf { i } + t \mathbf { j }$