Final answer:
Answer is D. √2Pt/m. The correct relationship between velocity (v) and time (t) for an automobile of mass m accelerating from rest with constant power p is v = √(2Pt/m).
Step-by-step explanation:
The original question seems to ask about how the velocity of an automobile varies with time if a constant power is supplied by the engine and the car starts from rest.
When a force generated from an engine does work, and this work changes the kinetic energy of the car, we can relate power, velocity, mass, and time together to find an equation that describes this relationship.
In Physics, power (P) is the rate at which work is done and is given by the formula P = dW/dt, where W is work and t is time.
The work done on the car is also equal to the change in kinetic energy, which is W = (1/2)m(v^2) - (1/2)m(0^2), since the car starts from rest.
As power is constant, integrating power with respect to time gives us the work done, W = Pt.
Therefore, Pt = (1/2)mv^2, and we solve for v, finding that v = √(2Pt/m).
So, the correct relationship between velocity and time when an automobile of mass m accelerates from rest with constant power p is D. v = √(2Pt/m).