Final answer:
The distance moved by a body along a straight line by a machine delivering constant power is proportional to t¹/² because kinetic energy increases with time and is related to the square of velocity, implying that distance increases with the square root of time.
Step-by-step explanation:
The question involves constant power and the distance moved by a body along a straight line. When dealing with constant power, the work done on a body (or the energy transferred to the body) is equal to the power multiplied by the time for which the power is applied. Since power is constant, the work done increases linearly with time.
From physics, we know that work is also the product of force and distance (W = Fd). With a constant force, this means that the distance moved will be proportional to the work done. However, when power is constant and the force varies (since force can increase as velocity increases), the relationship between distance and time must take into account the changing velocity.
For a machine delivering constant power, the kinetic energy (K) of the body is also increasing linearly with time since power is the rate of energy transfer. And, since kinetic energy is proportional to the square of the velocity (v), we can infer that velocity increases with the square root of time. Therefore, the distance (D) covered is related to the square of the velocity, which means that D is proportional to t¹/².