To find the force as a function of time, we utilize the relationship between power, work, and kinetic energy:

1. Work-Energy Theorem: Since the machine delivers constant power $k$, the work done ($W$) on the particle in time $t$ is $W = P \times t = kt$ . According to the work-energy theorem, this work is equal to the change in kinetic energy: $kt = \frac{1}{2}mv^2 - 0$ .
2. Velocity Calculation: From the energy equation, we solve for velocity ($v$): $v^2 = \frac{2kt}{m} \implies v = \sqrt{\frac{2kt}{m}}$ .
3. Force Calculation: Power is defined as the product of force and velocity ($P = F \cdot v$) . Therefore, $F = \frac{P}{v}$.
4. Substitution: Substituting the expressions for $P$ and $v$: $F = \frac{k}{\sqrt{\frac{2kt}{m}}} = \sqrt{\frac{k^2 \cdot m}{2kt}} = \sqrt{\frac{mk}{2}} t^{-1/2}$

Thus, the force acting on the particle at time $t$ is $\sqrt{\frac{mk}{2}} t^{-1/2}$.