(i) To find the speed of the sledge at the bottom of the slope, we use the conservation of energy. The potential energy (PE) lost is converted into kinetic energy (KE). The loss in potential energy is given by:

\(\text{PE loss} = mg \times 100 \sin 20^{\circ}\)

Using the equation for kinetic energy gain:

\(\frac{1}{2}mv^2 - \frac{1}{2}m \times 5^2 = mg \times 100 \sin 20^{\circ}\)

Solving for \(v\), we find:

\(v = 26.6 \text{ m/s}\)

(ii) When resistance is present, the work done against resistance is 8500 J. The kinetic energy at the bottom is given by:

\(\text{KE} = \frac{1}{2}m \times 21^2 - \frac{1}{2}m \times 5^2\)

\(\text{KE} = \pm (0.5m \times 441 - 0.5m \times 25) = \pm 208m\)

The equation for energy balance is:

\(mg \times 100 \sin 20^{\circ} = 8500 + 208m\)

Solving for \(m\), we find:

\(m = 63.4 \text{ kg}\)