Once I discussed cars and there was a question about rate at which the car gains its speed ($100\:\mathrm{km/h}$) from standstill (i.e. acceleration). At some point of conversation it came up that cars cannot gain their $100\:\mathrm{km/h}$ speed for such tiny time interval I proposed (less than $3$ seconds), because it can't go with acceleration more than $g$ i.e. free fall accel on Earth (we talked about wheel-driven vehicles - not a rocket-driven - and all we know that at certain acceleration value the wheels lose their grip with ground and vehicle is not moving forward it's burning tyres istead).
I was disagree, I feel intuitively that the critical acceleration for the wheel traction exists but it couldn't be equal to $g$ or approaching to, because it should look like a some deep law of the nature (between friction and acceleration) and this phenomenon would be widely known (but I couldn't find anything).
We started searching for contra-evidence. I found Bugatti Veyron which achieves $100\:\mathrm{km/h}$ speed roughly for $2.5\:s$ and some Formula 1 cars. But the sources of acceleration values were considered as non-reliable and not proven.
I know that the friction and cohesion proccesses are very untrivial, it brings electromagnetic interactions between atoms and molecules on microscopic level as well as surface geometry on macrolevel - that was another my argument against the statement that acceleration can't be more than $g$ without traction loss - because such complex and heterogeneous process as wheel traction can't have such a simple solution and equal to $g$.
So, could someone prove it or provide a formula or methods to show that critical acceleration of wheel-driven body without traction loss could be more than $g$ (or less)?
Answer
There is an approximate law that states that the frictional force, $F$, is given by:
$$ F = \mu W $$
where $W$ is the normal force and $\mu$ is the coefficient of friction; $\mu$ is normal taken to lie in the range zero to one. For a car the weight $W$ is given by $mg$, and the acceleration would be $F/m$, so dividing through by $m$ we get th acceleration to be:
$$ a = \mu g $$
If $\mu$ has the maximum value of one then the acceleration cannot exceed $g$ because trying to accelerate faster would just make the tyres spin.
But ...
The equation we started with is an approximation, and friction is actually a far more complicated phenomenon that that simple law suggests. For example car tyres deform and can key into irregularities on the road to increase the friction. Racing car tyres can have effective values of $\mu$ far greater than one, so they can accelerate at more than 1g. In fact a drag racer can accelerate at around 4g.
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