calculus homework

Question

The velocity function, in feet per second, is given for a particle moving along a straight line.

v(t) = t² − t − 72, 1 ≤ t ≤ 16

(a) Find the displacement.

(b) Find the total distance that the particle travels over the given interval.

Answer 1

t²-t-72 = (t-9)(t+8) =0 t = 9, -8. Ignore -8 since not in [1,16].

If you plot v(t), you will see that v(t) is negative from 1 to 9 and positive from 9 to 16.

So, the particle is backing up from 1 to 9 and going forward from 9 to 16.

The displacement = ∫₁¹⁶v(t)dt

The total distance traveled = -∫₁⁹v(t)dt+∫₉¹⁶v(t)dt