I have been studying how a the rotation and translation of a sliding disc (think of it as a hockey puck) is affected by uniform friction. I encountered an integral that I was not able to solve, and worked around it by generating artificial data with a computer model. There i found these limits for the torque $\tau$ on the disc:
In the limit of low $v$ (translational speed), the torque seems to be given by: $\tau = -\text{sign}(\omega)\left (e^{\left (\frac{|v|}{\omega} \right )^2}-\frac{5}{3} \right )$.
In the limit of high $v$, thet torque is given by: $\tau \approx -\frac{\omega}{5.65|v|}$.
Here is a plot that illustrates this behaviour:
In the picture above, the torque $\tau$ (Nm) measured in my computer model is plotted as a function of the translational speed $v$ (m/s). The angular velocity was here $\omega = 6.9697$ rad/s blue color is the measured data, red color is my curve fit in the limit of low $v$, green color is my curve fit in the limit of high $v$.
As can be seen, these limiting fits asypthotes the measured values from below. The measured values are not some weighted mean between the limiting cases, since my curve fits are both above the measured values for high $v$.
How can I join these limiting cases into a function with the correct behaviour in both the limits of high $v$ and of low $v$? Is there a general procedure for extrapolating limits as in this case, or do you have a creative idea?