To find the values of \(P\) and \(R\), we use the formula for power: \(P = Fv\), where \(F\) is the driving force and \(v\) is the velocity. The driving force \(F\) can be expressed as \(F = ma + R\), where \(m\) is the mass, \(a\) is the acceleration, and \(R\) is the resistance.

For the first condition (speed = 14 m s\(^{-1}\), acceleration = 1.4 m s\(^{-2}\)):

\(1000 \frac{P}{14} - R = 800 \times 1.4\)

For the second condition (speed = 25 m s\(^{-1}\), acceleration = 0.33 m s\(^{-2}\)):

\(1000 \frac{P}{25} - R = 800 \times 0.33\)

Solving these two equations simultaneously:

1. \(1000 \frac{P}{14} - R = 1120\)

2. \(1000 \frac{P}{25} - R = 264\)

Subtract equation 2 from equation 1:

\(1000 \left( \frac{P}{14} - \frac{P}{25} \right) = 1120 - 264\)

\(1000 \left( \frac{25P - 14P}{350} \right) = 856\)

\(1000 \frac{11P}{350} = 856\)

\(11P = 299.6\)

\(P = 27.2 \text{ kW}\)

Substitute \(P = 27.2\) back into one of the original equations to find \(R\):

\(1000 \frac{27.2}{14} - R = 1120\)

\(1942.857 - R = 1120\)

\(R = 1942.857 - 1120\)

\(R = 822.857 \approx 825 \text{ N}\)