# Why is the formula for maximum speed in frictionless surface used to calculate optimum speed to avoid wear and tear of tires?

In my school textbook , under the concept of friction I studied maximum possible velocity of a car while taking a turn in normal roads is given by: $${v}_{max}=\sqrt{{\mu}_{s}rg}$$ , and on banked road it is: $${v}_{max}=\sqrt{rg\times \frac{{\mu}_{s}+\tan\left(\theta \right)}{1-{\mu}_{s}\tan\left(\theta \right)}}$$ , where $${\mu}_{s}$$ is the static coefficient of friction, $$r$$ is the radius and $$g$$ is acceleration due to gravity.

And then I studied that when we ignore friction, the formula turns out to be $${v}_{max}=\sqrt{rg\tan\left(\theta \right)}$$, considering road can be banked too.

Now, I have seen when questions like these are posed:

What is the optimum speed of the car to avoid any wear and tear of tires? (Given some value of coefficient of friction)

The solutions use $${v}_{max}=\sqrt{rg\tan\left(\theta \right)}$$ ? So here's my question:

1. When its mentioned that friction is present, why do we just assume it friction less?
2. I have also studied that both area of contact and velocity have no relation to friction (at least in case of static,rolling and sliding friction), then also it doesn't make sense to evaluate some "optimum" speed at some friction?
• Thats how to ask a question on se bro Aug 7, 2021 at 9:25