A car initiates its motion from rest on a half kilometer long...

The maximum acceleration of the car is provided by the frictional force between the tyres and the road.

f_{\text\max} = \mu N = \mu mg

ma_{\text\max} = \mu mg \Rightarrow a_{\text\max} = \mu g

Substituting the given values $\mu = 1.0$ and taking $g = 10 \text{ m/s}^2$:

a_{\text\max} = 1.0 \times 10 = 10 \text{ m/s}^2

The car starts from rest, so the initial velocity $u = 0$. The length of the bridge is $s = 0.5 \text{ km} = 500 \text{ m}$.

Using the equation of kinematics to find the minimum time $t$:

s = ut + \dfrac{1}{2} a_{\text\max} t^2

$500 = 0 + \dfrac{1}{2} (10) t^2$

$5 t^2 = 500$

$t^2 = 100$

$t = 10 \text{ s}$