The position of an object moving on a straight line is defined by the relation `x=t^(3)-2t^(2)-4t`, where x is expressed in meter and t in second. Determine
(a) the average velocity during the interval of 1 second to 4 second.
(b) the velocity at t = 1 s and t = 4 s,
(c) the average acceleration during the interval of 1 second to 4 second.
(d) the acceleration at t = 4 s and

Explanation

(a) The position at `t=1s, x_(1)=1^(3)-2(1)=-5m`
The position at `t=4s, x_(2)=4^(3)-2(4)^(2)-4(4)=16m`
Average velocity in the interval `=(Deltax)/(Deltat)=(x_(2)-x_(1))/(Deltat)`
`=(16-(-5))/(4-1)`
`=(21)/(3)=7m//s`
(b) By the term velocity we mean instantaneous velocity which is given by
`v=(dx)/(dt)=3t^(2)-4t-4`
At `t=1s, v_(1)=3(1)^(2)-4(1)-4=-5m//s`
At `t=4 s, v_(2)=3(4)^(2)-4(4)-4=28m//s`
(c) The average acceleration in an interval is given by
`a_(avg)=(v_(2)-v_(1))/(Deltat)`
`=((28)-(-5))/(4-1)=(33)/(3)=11m//s^(2)`
(d) The instantaneous acceleration is given by
`a=(dv)/(dt)=6t-4`
At `t=4s, a=6(4)-4=20m//s^(2)`