(a) To find the initial velocity v, we use the equation of motion:

\(v = u + at\)

where \(v = 0\) (velocity at the greatest height), \(a = -g = -10 \text{ m/s}^2\) (acceleration due to gravity), and \(t = 3 \text{ s}\).

Substituting the values, we get:

\(0 = v - 10 \times 3\)

\(v = 30 \text{ m/s}\)

(b) To find the greatest height, we use the equation:

\(v^2 = u^2 + 2as\)

where \(v = 0\), \(u = 30 \text{ m/s}\), and \(a = -10 \text{ m/s}^2\).

Substituting the values, we get:

\(0 = 30^2 + 2(-10)s\)

\(0 = 900 - 20s\)

\(20s = 900\)

\(s = 45 \text{ m}\)

Thus, the greatest height is 45 m.