Final answer:
A particle moving in a circle with equal centripetal and tangential accelerations can theoretically do so for an infinite amount of time, provided there is a constant force to sustain the motion.
Step-by-step explanation:
The student's question pertains to the maximum time a particle can move in a circular path with its centripetal acceleration (ac) and tangential acceleration (at) being equal. When a particle moves in a circle of radius R with an initial velocity V₀, and both the centripetal and tangential accelerations are equal at all times, the particle is said to be in nonuniform circular motion because its speed is changing due to the tangential acceleration. The centripetal acceleration (ac) points towards the center of the circle and is calculated by the formula ac = v²/R, where v is the instantaneous tangential velocity of the particle.
For a particle executing this type of motion, there is no stated maximum time limit for which it can move. As long as the force providing the centripetal acceleration can continue to match the tangential acceleration this system is theoretically able to sustain its motion indefinitely. Thus, the maximum time for which a particle can move under these conditions is infinite.