Can someone explain this please?

Question

The acceleration function (in m/s²) and the initial velocity v(0) (in m/s) are given for a particle moving along a line.

a(t) = 2t + 2,    v(0) = −15,    0 ≤ t ≤ 5

I found the velocity (in m/s) at time t equation, which is v(t) = t^2 + 2t - 15

Now, how can I find the distance traveled (in m) during the given time interval.

Answer 1

I'm assuming by the question that you are looking fr the total distance traveled and not the displacement.

If you are doing this via a calculator, just integrate the absolute value of t² + 2t - 15 between 0 and 5

To do it manually, find the time when velocity changes from negative to positive and then break the integration into two parts.

First solve t² + 2t - 15 = 0

(t - 3)(t + 5) = 0

t = 3 and t = -5

Use t = 3 as your dividing point. The velocity between t = 0 and t = 3 is negative while the velocity between t = 3 and t = 5 is positive:

Total distance = |₀∫³ t² + 2t - 15 dt| + ₃∫⁵t² + 2t - 15 dt = |-27| + 56/3 = 137/3 meters