(a) The power of the winch is given as 4.5 kW, which is 4500 W. The work done by the winch is equal to the force times the distance, which is also equal to power times time. Therefore, we have:

\(4500 = \frac{600 \times 15}{t}\)

Solving for \(t\), we get:

\(t = \frac{600 \times 15}{4500} = 2 \text{ s}\)

(b) After the rope breaks, the only force acting on the crate is the resistance of 600 N. Using Newton's second law, \(F = ma\), we have:

\(600 = 200a\)

\(a = 3 \, \text{m/s}^2\)

The initial velocity \(u\) is the velocity at which the crate was moving before the rope broke. From part (a), the velocity \(v\) is:

\(600v = 4500 \Rightarrow v = 7.5 \, \text{m/s}\)

Using the equation \(v = u + at\) with \(v = 0\) (since the crate comes to rest), \(u = 7.5\), and \(a = -3\):

\(0 = 7.5 - 3t\)

Solving for \(t\), we get:

\(t = \frac{7.5}{3} = 2.5 \text{ s}\)