Using the conservation of momentum, the initial momentum of the system is given by:

\(0.1 imes 4 + 0 imes 0 = 0.4v + 0.1v\)

Solving for \(v\), we have:

\(0.4v + 0.1v = 0.4 \times v + 0.1 \times v = 0.4\)

Thus, \(0.5v = 0.4\)

\(v = \frac{0.4}{0.5} = 0.8 \text{ m/s}\)

Alternatively, considering the possibility of opposite direction:

\(0.1 imes 4 + 0 = 0.4v + 0.1(-v)\)

\(0.4 = 0.4v - 0.1v\)

\(0.4 = 0.3v\)

\(v = \frac{0.4}{0.3} = \frac{4}{3} \approx 1.33 \text{ m/s}\)

Therefore, the two possible speeds of P after the collision are 0.8 m/s and 1.33 m/s.