I got
this answer from stackexchange, but I had to fix and complete it due to several errors:
no air resistance:
$$a(v) =\frac F m = \frac {\frac P v } m = \frac P {vm} = \frac P m \frac 1 v = \frac w v$$
with air resistance:
$$a(v) = \frac{w}{v} - \frac 1 2 \rho C_d A v^2 = \frac{w}{v} - C_2 v^2$$
$$C_2= \frac 1 2 \rho C_d A \frac 1 m$$
$$w(v) = \frac {P(v)} m = \frac {vF(v)} m = \frac {v F_{max}(1-\frac{rpm(v)}{rpm_{max}})} m = \frac {v F_{max}(1-\frac{\gamma v}{rpm_{max}})} m = \frac {vF_{max}} m - \frac{\gamma v^2 F_{max}} {m \times rpm_{max}} = C_0v - C_1v^2$$
$$a(v) = \frac{w}{v} - C_2 v^2 $$
$$\boxed {a(v)= C_0 - C_1v - C_2 v^2} $$
##C_0=\frac {F_{max}} m ##
##C_1=\frac{\gamma F_{max}} {m \times rpm_{max}} ## Torque (and force) dependence from speed
##C_2= \frac 1 2 \rho C_d A \frac 1 m## Air drag
##F_{max} = \frac {T_{max}} {r_w}##
##r_w## = wheel radiusIf Torque is constant vs speed (as in electric cars):
$$a(v) = C_0 - C_2 v^2$$
(rolling friction not yet taken into account)
Without air friction (not what I am looking for, but useful for comparison and for the example):
$${ t_{60} = \frac{P_{max}}{2 m \epsilon^2 g^2} + \frac{m v_{60}^2}{2 P_{max}} } $$
Example
A ##m=1200\,{\rm kg}## car with peak power ##P_{max} = 160\,{\rm hp} = 119,000\,{\rm W}## goes to ##v_{60} = 26.9\,{\rm m/s}##. Traction is ##\epsilon=0.4## and ##g=9.81\,{\rm m/s^2}##
$$ t_{60} = \frac{ \frac{119,000}{1200} }{2 \times 0.4^2 \times9.81^2} + \frac{26.9^2}{2 \frac{119,000}{1200}} = 3.23 + 3.63 = 6.86 \, {\rm sec} $$
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Going on:
$$ a(v) = C_0 - C_1 v - C_2 v^2 $$
With direct integration you have
$$ t_1 = \int_0^{v_1} \frac{1}{a}\,{\rm d}v = \int_0^{v_1} \frac{1}{C_0-C_1 v-C_2 v^2}\,{\rm d}v = \ldots$$
With the parameter of top speed ##a(v_f) = 0 \ ## ==> ##v_f = \dfrac{\sqrt{C_1^2+4 C_0 C_2}-C_1}{2 C_2} ## and the dimensionless parameter ##\zeta = 2-\frac{C_1 v_f}{C_0}## the time to speed is
## t(v) = \frac{v_f}{C_0 \zeta} \ln \left(1+\zeta \frac{v}{v_f-v}\right) ## Variable torqueFor constant torque, ##C_1=0## and ##\zeta = 2## , hence:
$$t(v) = \frac{v_f}{2 C_0 } \ln \left(1+ \frac{2v}{v_f-v}\right) = \frac{v_f}{2 C_0 } \ln \left( \frac{v_f+v}{v_f-v}\right)$$
##\boxed{t(v) = \frac{v_f}{2 C_0 } \ln \left( \frac{v_f+v}{v_f-v}\right)}## Constant torque
##v_f = \dfrac{\sqrt{C_1^2+4 C_0 C_2}-C_1}{2 C_2} = \frac{\sqrt{4 C_0 C_2}}{2 C_2} ##
##v_f= \sqrt \frac {C_0}{C_2} ## AirDrag-limited top speed for constant torque$$\boxed{t_{100kmh}= \frac 1 {2 \sqrt{C_2C_0}} ln{ \left( \frac{\sqrt{\frac {C_0}{C_2}}+27.8} {\sqrt{\frac {C_0}{C_2}}-27.8 } \right)} }$$
$$\boxed{C_0=\frac {T_{max}} {rm} }$$
$$\boxed{C_2= \frac 1 2 \rho C_d A \frac 1 m}$$
T = torque (Nm)
r = wheel radius (m)
m = vehicle mass (kg)
##\rho## = air density = 1.225 kg/m3
##C_d## = air drag coefficient (dimensionless, around 0.30 for cars)
A = frontal area (m2), around 2,2 for cars
Note: later I'll double check steps...
