A body is moving in a straight line along x-axis. Its distance from the origin is given by the equation x=${{at}^{2}}$ - ${{bt}^{3}}$, where x is in metre and t is in second. Find

(i) the average speed of the body in the interval t = 0 and t = 2 s and

(ii) its instantaneous speed at t = 2s

The given equation x = ${{at}^{2}}$ —${{bt}^{3}}$

(i)If t = 0 ${ x }*{ 0 }$
If t= 2s ${ x }*{ 2 }$ = 4a - 8b

∆x = ${ x }

*{ 2 }$ -*

${ x }{ 0 }$ = 4a-8b = 0

${ x }

4a-8b

∆t = 2-0 = 2

Average speed in the given interval of time

${ v }_{ av }$ = ∆x / ∆t = 4a-8b / 2 = 2a-4b

(ii)Instantaneous speed

v =dx/dt = d/dt (${{at}^{2}}$ - ${{bt}^{3}}$)

= 2at - 3${{bt}^{2}}$

At t=2 sec , v = 4a- 12 b m/s