If this is the case, the equation either needs a domain readjustment(so, valid from V∈[0.7-∞]) or an equation redefinition.

Using a piecewise function would look like: 
$$
M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), V∈[0.7-∞]
$$
$$
M \frac{dv}{dt} = 0, V∈(0-0.7)
$$

Friction, is just a reactionary force. For rolling resistance can be modelled as an example of a dynamic friction. [Hence, until motion occurs][1], $$F_t = β_1$$
Then, you should get better looking profiles.


  [1]: https://www.researchgate.net/figure/Static-and-dynamic-friction-behaviour-70_fig3_326623707