Find the velocity of a particle that has the given acceleration and the given initial velocity. $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$
A) $\mathbf { v } ( t ) = 12 \mathbf { i } - \mathbf { j } + 3 t \mathbf { k }$
B) $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + t \mathbf { k }$
C) $\mathbf { v } ( t ) = ( 12 t - 9 ) \mathbf { i } + 7 t \mathbf { k }$
D) $\mathbf { v } ( t ) = \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$
E) $\mathbf { v } ( t ) = 12 \mathbf { i } - 7 \mathbf { j } + 3 t \mathbf { k }$

Question

Quizplus · Accepted Answer

Using the given acceleration, we can integrate it twice to find the velocity function:
\begin{align*}
\mathbf{v}(t) &= \int \mathbf{a}(t) dt \
&= \int 3\mathbf{k} dt \
&= 3t\mathbf{k} + \mathbf{C}_1
\end{align*}
where $\mathbf{C}_1$ is a constant vector of integration. 
Next, we use the given initial velocity to solve for $\mathbf{C}_1$:
\begin{align*}
\mathbf{v}(0) &= 3(0)\mathbf{k} + \mathbf{C}_1 \
\mathbf{C}_1 &= \mathbf{v}(0) \
&= 12\mathbf{i} - 7\mathbf{j}
\end{align*}
Thus, the velocity function is:
\begin{align*}
\mathbf{v}(t) &= 3t\mathbf{k} + (12\mathbf{i} - 7\mathbf{j}) \
&= \boxed{12\mathbf{i} - 7\mathbf{j} + 3t\mathbf{k}}
\end{align*}

Find the Velocity of a Particle That Has the Given a(t)=3k,v(0)=12i−7j\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }a(t)=3k,v(0)=12i−7j

Find the Velocity of a Particle That Has the Given $\mathbf { a } ( t ) = 3 \mathbf { k } , \mathbf { v } ( 0 ) = 12 \mathbf { i } - 7 \mathbf { j }$