Question

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. What is the retardation of the particle when its volume become zero?

Answer 1

The formula of retardation is just opposite of acceleration. You can find by using formula
V-U/t

Answer 2

$\text{displacement,} \: x = 8 + 12t - t^3 \\ \\ \text{Velocity,}\ v = \frac{dx}{dt}= \frac{d}{dt} \big(8+12t-t^3 \big) = 12-3t^2\\ \\ \text{acceleration,}\ a= \frac{dv}{dt}= \frac{d}{dt} \big(12-3t^2 \big)=-6t\\ \\ \text{When velocity becomes 0}\\ \\ \Rightarrow v=0\\ \\ \Rightarrow 12-3t^2=0\\ \\ \Rightarrow 3t^2=12\\ \\ \Rightarrow t^2=\frac{12}{3}=4\\ \\ \Rightarrow t=\sqrt{4}=2\ seconds\\ \\ \text{acceleration at that time is given as}\\ \\ a_{(t=2)}=-6 \times 2 = -12\ ms^{-2}\\ \\ \text{Required retardation is 12 ms {}^{-2} }$