Final answer:
The velocity of a particle as a function of time, when acted on only by air resistance, can be found analytically using Newton's second law and integrating the acceleration equation. The final expression for the velocity takes into account the initial velocity and time.
Step-by-step explanation:
The velocity of a particle as a function of time, when acted on only by air resistance, can be found analytically. The force of air resistance is given by Fp = -bx², where b is a constant. We can use Newton's second law, F = ma, to relate the force, mass, and acceleration of the particle. In this case, the acceleration is given by a = Fp / mp = -bx² / mp.
To find the velocity as a function of time, we integrate the acceleration with respect to time. This gives us the expression v(t) = -b/3m * t³ + C, where C is a constant of integration. To determine the value of C, we use the initial condition v(t₀) = v₀, where v₀ is the initial velocity and t₀ is the initial time. Substituting these values into the equation, we can solve for C and obtain the final expression for the velocity.
Therefore, the velocity of the particle as a function of time is v(t) = -b/3m * t³ + (v₀ + b/3m * t₀³).