To meet a U.S. Postal Service requirement, employees' footwear must have a coefficient of static friction of 0.5 or more on a specified tile surface. A typical athletic shoe has a coefficient of 0.870. In an emergency, what is the minimum time interval in which a person starting from rest can move 3.20 m on a tile surface if she is wearing the following footwear?

(a) footwear meeting the Postal Service minimum
(b) a typical athletic shoe

To determine the minimum time interval required to move a distance of 3.20 m on a tile surface, we need to use the coefficients of static friction for the given footwear.

(a) Footwear meeting the Postal Service minimum:
The minimum coefficient of static friction required by the U.S. Postal Service is 0.5. To find the minimum time interval, we can use the equation:

\( \text{Distance (d)} = \frac{1}{2} \times \text{Coefficient of static friction} \times \text{Acceleration (a)} \times \text{Time interval (t)}^2 \)

Rearranging the equation, we get:

\( t = \sqrt{\frac{2d}{\text{Coefficient of static friction} \times \text{Acceleration}}}\)

Plugging in the given values:
\(d = 3.20\) m
\(\text{Coefficient of static friction} = 0.5\)
\(\text{Acceleration} = 9.8\) m/s² (assuming acceleration due to gravity)

\(t = \sqrt{\frac{2 \times 3.20}{0.5 \times 9.8}}\)

Calculating this, we find:
\(t \approx 0.8\) seconds

Therefore, the minimum time interval required to move 3.20 m on a tile surface with footwear meeting the Postal Service minimum is approximately 0.8 seconds.

(b) Typical athletic shoe:
The typical athletic shoe has a coefficient of static friction of 0.870. We can follow the same process as above, using this coefficient instead:

\(t = \sqrt{\frac{2 \times 3.20}{0.870 \times 9.8}}\)

Calculating this, we find:
\(t \approx 0.54\) seconds

Therefore, the minimum time interval required to move 3.20 m on a tile surface with a typical athletic shoe is approximately 0.54 seconds.

Note: These calculations assume no additional external forces or changes in the coefficient of static friction during the motion.