(a) If a submarine’s sonar can measure echo times with a precision of \(0.0100\;s\), what is the smallest difference in distances it can detect? (Assume that the submarine is in the ocean, not in fresh water.) (b) Discuss the limits this time resolution imposes on the ability of the sonar system to detect the size and shape of the object creating the echo.

Because sound waves are reflected, an echo is a sound that is repeated. Like a rubber ball bounces off the ground, sound waves can bounce off smooth, complicated things. Even though the sound's direction changes, the echo sounds identical to the original.

The time of travel is \(T = 0.0100\;{\rm{s}}\).

The speed of sound in ocean water is \(v = 1540\;{\rm{m/s}}\).

The distance of the object creating echo is given by,

\(d = 0.5vT\)

Plugging the values,

\(\begin{align}d &= 0.5 \times 1540 \times 0.0100\\ &= 7.7\;{\rm{m}}\end{align}\)

Hence, the required distance is \(7.7\;{\rm{m}}\).

The resolution imposes on the sonar to detect the size is that the sonar will be able to see the large objects, not the more minor things, but their shape and size. We need different arrangements for that.