  We apply a recent formalism of quantum geodesics to the well-known
bicrossproduct model $\lambda$-Minkowski quantum spacetime
$[x^i,t]=\imath\lambda_p x^i$ with its flat quantum metric as a model of
quantum gravity effects, with $\lambda_p$ the Planck scale. As examples,
quantum geodesic flow of a plane wave gets an order $\lambda_p$ frequency
dependent correction to the classical geodesic velocity. A quantum geodesic
flow with classical velocity $v$ of a Gaussian with width $\sqrt{2\beta}$
initially centred at the origin changes its shape but its centre of mass moves
with ${<x>\over<t>}=v(1+{\lambda_p^2\over 2\beta}+O(\lambda^3_p))$, an order
$\lambda_p^2$ correction. This implies, at least within perturbation theory,
that a `point particle' cannot be modelled as an infinitely sharp Gaussian due
to quantum gravity corrections. For contrast, we also look at quantum geodesics
on the noncommutative torus with a 2D curved weak quantum Levi-Civita
connection.
