The acceleration of the particle down the plane is given by:

\(a = g \sin 30^{\circ}\)

where \(g\) is the acceleration due to gravity \((9.8 \text{ m/s}^2)\).

Thus, \(a = 9.8 \times 0.5 = 4.9 \text{ m/s}^2\).

For part (i):

Using the equation \(v^2 = u^2 + 2as\), where \(u = 0\), \(s = 0.9 \text{ m}\), and \(a = 4.9 \text{ m/s}^2\):

\(v^2 = 2 \times 4.9 \times 0.9\)

\(v^2 = 8.82\)

\(v = \sqrt{8.82} \approx 2.97 \text{ m/s}\)

Rounding to the nearest whole number, the speed is 3 m/s.

For part (ii):

Using the equation \(v = u + at\), where \(u = 0\), \(a = 4.9 \text{ m/s}^2\), and \(t = 0.8 \text{ s}\):

\(v = 0 + 4.9 \times 0.8\)

\(v = 3.92 \text{ m/s}\)

Rounding to the nearest whole number, the speed is 4 m/s.