Final answer:
The speed of the particle is proportional to the square root of time (a. √t) when the work done by a net force is proportional to time, due to the linear relationship between work and kinetic energy. So the correct option is a.
Step-by-step explanation:
The student has asked about the relationship between the work done by a net force on a particle and the resulting speed of that particle, specifically when the work rate is proportional to time. The question at hand is to determine how the speed of the particle is proportional to time t.
When the work done on a particle is proportional to time t, the power, which is the rate of work done, is also proportional to time. Power is defined as P = dW/dt, and since W = Fd (where F is force and d is displacement), we can infer that the force exerted is constant. Now, recalling that work done is related to kinetic energy (K), we have that K = (1/2)mv^2. Therefore, as the work done increases linearly with time due to power being proportional to time, kinetic energy and thus the square of the speed must also increase linearly. Thus, v∝√t. Therefore, the speed of the particle is proportional to the square root of time.
So the answer is: a. √t