A force with magnitude $14 \mathrm {~N}$ acts directly upward from the xy-plane on an object with mass 6 $\mathrm { kg }$. The object starts at the origin with initial velocity $\mathbf { v } ( 0 ) = 7 \mathbf { i } - 2 \mathbf { j }$. Find its position function.
Select the correct answer.
A) $\mathbf { r } ( t ) = 7 t \mathbf { i } - 2 t \mathbf { j }$
B) $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } - 7 t ^ { 2 } \mathbf { j } + 14 t ^ { 3 } \mathbf { k }$
C) $\mathbf { r } ( t ) = 2 t \mathbf { i } - 4 t \mathbf { j } + t ^ { 2 } \mathbf { k }$
D) $\mathbf { r } ( t ) = 5 t ^ { 3 } \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$
E) $\mathbf { r } ( t ) = 7 t \mathbf { i } - 2 t \mathbf { j } + \frac { 7 } { 6 } t ^ { 2 } \mathbf { k }$

Question

Quizplus · Accepted Answer

Since there is only one force acting on the object, we can use Newton's second law to find the acceleration:

\begin{align*}
\mathbf { F } & = m \mathbf { a } \
14 \mathbf { k } & = 6 \mathbf { a } \
\mathbf { a } & = \frac { 7 } { 3 } \mathbf { k }
\end{align*}

Using the initial conditions, we can find the velocity by integrating the acceleration:

\begin{align*}
\mathbf { v } ( t ) & = \mathbf { v } ( 0 ) + \int _ { 0 } ^ { t } \mathbf { a } ( t ) dt \
& = 7 \mathbf { i } - 2 \mathbf { j } + \frac { 7 } { 3 } t \mathbf { k }
\end{align*}

Finally, we integrate the velocity to get the position function:

\begin{align*}
\mathbf { r } ( t ) & = \mathbf { r } ( 0 ) + \int _ { 0 } ^ { t } \mathbf { v } ( t ) dt \
& = 0 + \int _ { 0 } ^ { t } \left( 7 \mathbf { i } - 2 \mathbf { j } + \frac { 7 } { 3 } t \mathbf { k } ight) dt \
& = \left[ 7 t \mathbf { i } - 2 t \mathbf { j } + \frac { 7 } { 6 } t ^ { 2 } \mathbf { k } ight] _ { 0 } ^ { t } \
& = 7 t \mathbf { i } - 2 t \mathbf { j } + \frac { 7 } { 6 } t ^ { 2 } \mathbf { k }
\end{align*}

Therefore, the answer is choice E.

A Force with Magnitude 14 N14 \mathrm {~N}14 N Acts Directly Upward from the Xy-Plane on an Object with from the Xy-Plane

A Force with Magnitude $14 \mathrm {~N}$ Acts Directly Upward from the Xy-Plane on an Object with from the Xy-Plane