29. The motion of a particle along a straight line is described by equation : x = 8 + 12t − t3 where x is in metre and t in second. The retardation of the particle when its velocity becomes zero, is:

29. The motion of a particle along a straight line is described by equation \(x = 8 + 12t − t^3\) where x is in metre and t in second. The retardation of the particle when its velocity becomes zero, is:

(a)  24  \(ms^{−2}\)

(b) 0  \(ms^{−2}\)

(c) 6  \(ms^{−2}\)

(d) 12  \(ms^{−2}\)

Answer :  (d) 12  \(ms^{−2}\)

\( x = 8 + 12t - t^3 \)

\( v = \)\(\frac{dx}{dt}\)\( = 12 - 3t^2 \)

\( 12 - 3t^2 = 0 \)

\( \Rightarrow t = 2 \, \text{s} \)

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

\( a_{t=2} = -6(2) = -12 \, \text{m/s}^2 \)