A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in ms$^{-1})$ of the particle is zero, the speed (in ms$^{-1})$ after 5 s is ____.

Step 1: Identify the given parameters.

The given values for the particle are:

$$m = 0.2 \text{ kg}$$ $$P = 0.5 \text{ W}$$ $$u = 0 \text{ m s}^{-1} \text{ at } t = 0$$

Step 2: Relate power to kinetic energy.

Power is defined as the rate of change of kinetic energy.

$$P = \frac{dK}{dt}$$ $$dK = P \, dt$$

Step 3: Integrate to find the kinetic energy at $t = 5\text{ s}$.

Integrating both sides from $t = 0$ to $t = 5\text{ s}$ yields:

$$K = \int_{0}^{5} P \, dt$$ $$K = P t$$ $$K = 0.5 \times 5 = 2.5 \text{ J}$$

Step 4: Calculate the final speed of the particle.

Use the relation between kinetic energy and speed.

$$K = \frac{1}{2} m v^2$$ $$v = \sqrt{\frac{2K}{m}}$$ $$v = \sqrt{\frac{2 \times 2.5}{0.2}}$$ $$v = \sqrt{25} = 5 \text{ m s}^{-1}$$

Answer: 5