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: $$ M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0.7-∞] $$ $$ M \frac{dv}{dt} = 0, t∈(0-0.7) $$

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.

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

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

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.

Another way to do this would be to have a criteria, where the reactionary $β_1$ is just as big as $F_t$.

Therefore, $$ M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0,∞] $$ Where $$ \beta_1 =F_t, F_t∈[0,\beta_{max}] $$ $$ \beta_1 =\beta_{max}, F_t∈[\beta_{max},∞] $$

Here, $\beta_{max} $ is your initial value of $\beta_1$

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.

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: $$ M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0.7-∞] $$ $$ M \frac{dv}{dt} = 0, t∈(0-0.7) $$

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.