Find the displacement and distance traveled

Question

The velocity function (in meters per second) is given for a particle moving along a line.

v(t) = t² − 4t − 12,    1 ≤ t ≤ 8

(a) Find the displacement.

 __________ m

(b) Find the distance traveled by the particle during the given time interval.

 __________ m

Answer 1

v(t) = t^2 -4t -12 1<t<8

s(t) = the integral of v(t) = t^3/3 - 2t^2 -12t

evaluated between t=1 and t= 8

= (1/3)(8^3) - 2(8^2) -12(8) -[1/3 -2 -12] = 512/3 -128 - 96 - 1/3 +14

= 512/3 - 210 -1/3

= 512/3 - 631/3

= -119/3

= -39 2/3 meters

it traveled 39 2/3 meters

s(t) = t^3/3-2t^2 -12t +c, where c = an unknown constant. it cancels out when the integral is evaluated between limits

s(t) = t(t^2/3 -2t -12)

s(8) = 8(64/3-16-12) =8(64/3 -84/3) = 8(-20/3)= -160/3 = -53 1/3

s(1) = (1)(1/3-2-12) =-14 + 1/3 = -13 2/3

s(8)-s(1) = -53 1/3-(-13 2/3) = -53 1/3 + 13 2/3 = - 39 2/3