To determine the speed of a particle after $ t = 5 $ seconds, given a constant power, the following steps should be taken :

Power Equation: Power is the rate at which work is done, and work is the change in kinetic energy :

$$ P = \frac{d(W)}{dt} = \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) $$

Given Data :

Mass of the particle ($ m $) = 0.2 kg

Constant power ($ P $) = 0.5 W

Initial speed ($ v_0 $) = 0 m/s

Differentiating Kinetic Energy with Respect to Time :

$$ P = \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) = m v \frac{dv}{dt} $$

Rearranging to Solve for $ v \frac{dv}{dt} $ :

$$ 0.5 = 0.2 \cdot v \cdot \frac{dv}{dt} $$

$$ \frac{dv}{dt} = \frac{5}{2v} $$

Integrating Both Sides :

$$ v \, dv = \frac{5}{2} \, dt $$

$$ \int v \, dv = \int \frac{5}{2} \, dt $$

$$ \frac{v^2}{2} = \frac{5t}{2} + C $$

Applying Initial Condition (when $ t = 0 $, $ v = 0 $) :

$$ 0 = \frac{5 \cdot 0}{2} + C $$

$$ C = 0 $$

Solving for $ v $ :

$$ \frac{v^2}{2} = \frac{5t}{2} $$

$$ v^2 = 5t $$

$$ v = \sqrt{5t} $$

Speed After 5 Seconds :

$$ v = \sqrt{5 \cdot 5} = \sqrt{25} = 5 \, \text{m/s} $$

Thus, the speed of the particle after 5 seconds is 5 m/s.