# Thread: Integral question w/ velocity and acceleration

1. ## Integral question w/ velocity and acceleration

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

a(t) = 2t + 5. 0 < t<3
v(0) = -6

Find the velocity v(t) at time (t).

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

a(t) = 2t + 5. 0 < t<3
v(0) = -6

*note: It's less than or equal to, not just less than

Find the total distance d traveled during the time interval.

My question is once you integrate the expression, do you just solve what t is? Or is there more to it?

2. The distance traveled over a time period is the integral of the speed

which is |v(t)|

the velocity v(t) is the integral of the acceleration.

Let's do it

a(t) = 2t + 5

so v(t) = t^2 + 5t + C

v(0) = -6 = C

So v(t) = t^2 + 5t - 6

We want the distance traveled between t= 0 and t =3

See the attachment note v(t) < 0 if t <1 so |v(t)| = -v(t) 0<t<1

and v(t) > 0 for t >1 so |v| = v(t) 1 < t < 3