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, $$F_t = β_1$$ Then, you should get better looking profiles.