The acceleration function $($in m$/$s$2)$ and the initial velocity v$(0)$ are given for a particle moving along a line.

a$($t$)=2$t $+4,$ v$(0)=12,0$ t $6$

Exercise $($a$)$

Find the velocity at time t$.$

Step $1$

The velocity function is the antiderivative of the acceleration.

v$($t$)=$

$(2$t $+4)$ dt

$=$

$$$2$t$22+4$t

$+$ C

Step $2$

We must determine the value of C$.$ We know that

v$(0)=12.$

Substituting $0$ into our antiderivative gives $12=$ v$(0)=0$

$0$

$+$ C$.$

Therefore,

C $=-12$

$-12$

$.$

Step $3$

Therefore, what is the velocity function at time t$?$

v$($t$)=$

m$/$s