The velocity of a particle moving in a straight line is given by $v = t \left( t ^ { 2 } + 6 \right) ^ { 4 } + 3 t$ .
Find an expression for the position s after time t.

A) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + 3 t ^ { 2 } + C$
B) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 5 } + \frac { 3 t ^ { 2 } } { 2 } + C$
C) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
D) $s ( t ) = \frac { t \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
E) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t } { 2 } + C$

Question

The velocity of a particle moving in a straight line is given by $v = t \left( t ^ { 2 } + 6 \right) ^ { 4 } + 3 t$ .  
Find an expression for the position s after time t.
 
A) $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + 3 t ^ { 2 } + C$
B)   $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 5 } + \frac { 3 t ^ { 2 } } { 2 } + C$
C)   $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
D)   $s ( t ) = \frac { t \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t ^ { 2 } } { 2 } + C$
E)   $s ( t ) = \frac { \left( t ^ { 2 } + 6 \right) ^ { 5 } } { 10 } + \frac { 3 t } { 2 } + C$

Quizplus · Accepted Answer

To find the position $s(t)$ from the velocity $v(t) = t(t^2 + 6)^4 + 3t$, we integrate $v(t)$ with respect to $t$. The integral of $t(t^2 + 6)^4$ is $\frac{(t^2 + 6)^5}{10}$ and the integral of $3t$ is $\frac{3t^2}{2}$. Adding the constant of integration $C$ gives the correct expression for $s(t)$.

The Velocity of a Particle Moving in a Straight Line v=t(t2+6)4+3tv = t \left( t ^ { 2 } + 6 \right) ^ { 4 } + 3 tv=t(t2+6)4+3t

The Velocity of a Particle Moving in a Straight Line $v = t \left( t ^ { 2 } + 6 \right) ^ { 4 } + 3 t$