21. The position x of a particle with respect to time t along x-axis is given by x = 9t2 − t3 where x is in metres and t in second. What will be the position of this particle when it achieves maximum speed along the +ve x direction?

21. The position x of a particle with respect to time t along x-axis is given by \(x = 9t^2 − t^3\) where x is in metres and t in second. What will be the position of this particle when it achieves maximum speed along the +ve x direction?

(a) 54 m

(b) 81 m

(c) 24 m

(d) 32 m

Answer :  (a) 54 m

\( x = 9 t^2 - t^3 \)

\( v = \)\( \frac{dx}{dt} \)\( = 18 t - 3 t^2 \)

\( a = \)\( \frac{dv}{dt} \)\( = 18 - 6 t \)

\( a = 0 \)

\( \rightarrow 18 - 6 t= 0 \)

\( 6 t = 18 \)

\( t = \)\( \frac{18}{6} \)\( = 3 s \)

\( x_{( a = 0 )} = 9 (3)^2 - (3)^3 \)

\( = 54 m  \)