To solve this problem, we use the equations of motion and Newton's second law.

(i) Finding the tension in the string:

First, we find the acceleration of the system. Using the equation of motion:

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

where \(v = 0.6 \text{ m/s}\), \(u = 0 \text{ m/s}\), and \(s = 0.8 \text{ m}\).

Substituting the values, we get:

\(0.6^2 = 0 + 2a \times 0.8\)

\(a = 0.225 \text{ m/s}^2\)

Now, applying Newton's second law to the 0.3 kg particle:

\(T - 0.3g = 0.3a\)

Substituting \(g = 9.8 \text{ m/s}^2\) and \(a = 0.225 \text{ m/s}^2\):

\(T - 0.3 \times 9.8 = 0.3 \times 0.225\)

\(T = 3.07 \text{ N}\)

(ii) Finding the value of \(m\):

Applying Newton's second law to the \(m\) kg particle:

\(mg - T = ma\)

Substituting \(T = 3.07 \text{ N}\) and \(a = 0.225 \text{ m/s}^2\):

\(mg - 3.07 = m \times 0.225\)

Solving for \(m\):

\(m(9.8 - 0.225) = 3.07\)

\(m = 0.314 \text{ kg}\)