(a) When is the object moving backward and when is it moving forward? (b) Using (a), find the total distance travelled during the first 10 seconds.

Question

An object is moving with velocity v(t) = 4t − 12 in feet per second.

(a) When is the object moving backward and when is it moving forward?



(b) Using (a), find the total distance travelled during the first 10 seconds.

Note: If I walk 10 steps forward and 3 steps backward, the total distance I travelled is 13 steps(not 7 steps)

Answer 1

v(t) = 4t -12 = velocity in feet per second

the object is moving forward when 4t-12 >0

4t>12

t >12/4

t> 3 seconds

After 3 seconds it's moving forward, before 3 seconds,it's moving backwards

from time 0<t<3, it's moving backwards

position is the integral of v(t) =s(t)=2t^2 -12t+c

it's at c initially

at time t=3, it's at 18-36+c=c-18

in 3 seconds it moved from c to c-18 = 18 feet backwards

in 10 seconds it moved to 2(10^2) -12(10) +c =200-120+c = 80+c

from c to c-18 to 80+c =18 +18+80= 116 feet = total distance (back and forward) moved in 10 seconds

but just 80 feet displacement from its initial position, from c to 80+c