(i) The increase in kinetic energy \(\Delta KE\) is given by:

\(\Delta KE = \frac{1}{2} m v^2 - \frac{1}{2} m u^2\)

Since the crate starts from rest, \(u = 0\), and \(v = 2\) m/s:

\(\Delta KE = \frac{1}{2} \times 50 \times 2^2 = 100 \text{ J}\)

(ii) The work done against the resistance \(W_R\) is:

\(W_R = \text{Resistance} \times \text{Distance} = 250 \times 20 = 5000 \text{ J}\)

(iii) The work done by the force \(W_F\) is:

\(W_F = F \cdot d \cdot \cos \alpha = 400 \times 20 \times \cos \alpha\)

Using the work-energy principle, the net work done is equal to the change in kinetic energy:

\(W_F - W_R = \Delta KE\)

\(400 \times 20 \times \cos \alpha - 5000 = 100\)

\(8000 \cos \alpha = 5100\)

\(\cos \alpha = \frac{5100}{8000} = 0.6375\)

\(\alpha = \cos^{-1}(0.6375) \approx 50.4^\circ\)