A particle starts from rest and moves through a distance \(S = 12t^{2} - 2t^{3}\) metres in time t seconds. Find its acceleration in 1 second.

The distance moved by a particle is given by the equation \(S = 12t^{2} - 2t^{3}\), where S is the distance travelled in metres, and t is the time taken in seconds. To find the acceleration of the particle in 1 second, we need to differentiate the equation for distance with respect to time twice to obtain the equation for acceleration. First, we differentiate S with respect to time to get the velocity equation: $$\frac{dS}{dt} = \frac{d}{dt}(12t^{2} - 2t^{3}) = 24t - 6t^{2}$$ Next, we differentiate the velocity equation with respect to time to get the acceleration equation: $$\frac{d^2S}{dt^2} = \frac{d}{dt}(24t - 6t^{2}) = 24 - 12t$$ To find the acceleration in 1 second, we substitute t = 1 into the acceleration equation: $$\frac{d^2S}{dt^2} \bigg\rvert_{t=1} = 24 - 12(1) = 12ms^{-2}$$ Therefore, the acceleration of the particle in 1 second is 12\(ms^{-2}\).