How do you tune shock absorbers to best deal with a bumpy road? Remember critical damping, b^{2} = 4mk, is the smallest damping for which
oscillations don't occur. That might seem the best choice, but in fact slightly smaller damping, with a little oscillation, has a faster
response. We plot two curves corresponding to b = 0.5, the critical value for the k, m values we've given, and b = 0.4. We plot the areas
under the two curves (counting area below as positive as well) to give some idea of the amount of deviation from equilibrium. The area is
less for the subcritical value. In fact, it's almost the same for b = 0.2, but then looking at the curve it's not clear that this is a
comfortable solution! We also allow the option of starting at a given displacement at zero velocity (the default) and starting at the
equilibrium position but with some velocity (a sudden kick). For that situation, it seems critical value might be a little better. Both
scenarios no doubt occur in a bumpy car ride. Check out the walk-through below to learn more about critical damping of oscillators.

Walk-Through

How do you think this will change for m = 0.5 and k = 0.5?

Try it! Then try both equal to 1. How, exactly, has the critical damping changed?

What happens at m = 1 and k = 0.25? Check it out. How does this critical damping differ from m = 0.5 and k = 0.5? Or does it?