Step-by-step derivation:

1. Identify the given information:

   • Mass of the particle, m = 0.2 kg.
   • Constant power delivered, P = 0.5 W.
   • Initial speed of the particle, u = 0 m/s.
   • Time interval, t = 5 s.

2. Relate Power, Work, and Time: Power is defined as the rate at which work is done. For a constant power P, the work W done over a time interval t is given by: $W = P \times t$

3. Calculate the Work Done: Substituting the given values into the equation: $W = 0.5 \text{ W} \times 5 \text{ s} = 2.5 \text{ J}$ So, the total work done on the particle in 5 seconds is 2.5 Joules.

4. Apply the Work-Energy Theorem: The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy (ΔK). $W = \Delta K = K_f - K_i$ Where $K_f$ is the final kinetic energy and $K_i$ is the initial kinetic energy.

5. Calculate the Initial and Final Kinetic Energies:

   • The initial kinetic energy $K_i$ is: $K_i = \frac{1}{2} m u^2$ Since the initial speed u = 0, we have $K_i = 0$.
   • The final kinetic energy $K_f$ is: $K_f = \frac{1}{2} m v^2$ where v is the final speed we need to find.

6. Solve for the Final Speed (v): According to the Work-Energy Theorem: $W = K_f - K_i$ $2.5 \text{ J} = \frac{1}{2} m v^2 - 0$ Substitute the value of mass m = 0.2 kg: $2.5 = \frac{1}{2} (0.2) v^2$ $2.5 = 0.1 v^2$ Now, solve for $v^2$: $v^2 = \frac{2.5}{0.1} = 25$ Take the square root to find v: $v = \sqrt{25} = 5 \text{ m/s}$ Since speed is a non-negative quantity, we take the positive root.

Final Answer:

The speed of the particle after 5 s is 5 m/s.