  We study the motion of a one-dimensional particle which reverses its
direction of acceleration stochastically. We focus on two contrasting
scenarios, where the waiting-times between two consecutive acceleration
reversals are drawn from (i) an exponential distribution and (ii) a power-law
distribution $\rho(\tau)\sim \tau^{-(1+\alpha)}$. We compute the mean, variance
and short-time distribution of the position $x(t)$ using a trajectory-based
approach. We show that, while for the exponential waiting-time, $\langle
x^2(t)\rangle\sim t^3$ at long times, for the power-law case, a non-trivial
algebraic growth $\langle x^2(t)\rangle \sim t^{2\phi(\alpha)}$ emerges, where
$\phi(\alpha)=2$, $(5-\alpha)/2,$ and $3/2$ for $\alpha<1,~1<\alpha\leq 2$ and
$\alpha>2$, respectively.
  Interestingly, we find that the long-time position distribution in case (ii)
is a function of the scaled variable $x/t^{\phi(\alpha)}$ with an
$\alpha$-dependent scaling function, which has qualitatively very different
shapes for $\alpha<1$ and $\alpha>1$. In contrast, for case (i), the typical
long-time fluctuations of position are Gaussian.
