The position of a particle varies with time according to the relation `x=3t^(2)+5t^(3)+7t`, where x is in m and t is in s. Find
(i) Displacement during time interval t = 1 s to t = 3 s.
(ii) Average velocity during time interval 0 - 5 s.
(iii) Instantaneous velocity at t = 0 and t = 5 s.
(iv) Average acceleration during time interval 0 - 5 s.
(v) Acceleration at t = 0 and t = 5 s.

Explanation

`x=3t^(2)+5t^(3)+7t`
(i) `x_(1)=3(1)^(2)+5(1)^(3)+7xx1=15" "Deltax=x_(3)-x_(1)=168m`
`x_(3)=3(3)^(2)+5(3)^(3)+7xx3=183`
(ii) `v_(av)=(Deltax)/(Deltat)=(x_(5)-x_(0))/(5-0)=(3(5)^(2)+5(5)^(3)+7xx5)/(5-0)=147m//s`
(iii) `V=(dx)/(dt)=6t+15t^(2)+7" "V_(0)=7m//s" "V_(5)=412m//s`
(iv) `a_(av)=(DeltaV)/(Deltat)=(V_(5)-V_(0))/(5-0)=(412-7)/(5)=81m//s" Note : "a_(av) ne("Avg. velocity")/("time")`
(v) `a=(dv)/(dt)=6+30t`
(v) `a=(dv)/(dt)=6+30t`
`a_(0)=6m//s^(2), a_(5)=156m//s^(2)`