For a body moving along a straight line, the displacement as a function of time is given by
x(t)=3t2−12t+5
where x is in metres and t in seconds, for the time interval 0≤t≤6s.
(i) Derive an expression for the instantaneous velocity v(t) as a function of time.
(ii) Calculate the displacement at t=1s and t=5s, and hence determine the average velocity between these two instants.
(iii) Find the instantaneous velocity at t=3s (the mid-point of the interval considered in part (ii)).
(iv) Examine whether the instantaneous velocity at the mid-point equals the average velocity over the interval 1s to 5s, and justify your conclusion mathematically.