A constant power machine pulls a block on a smooth horizontal surface. Which one of the following correctly describes the relation between speed of the block (v) and time (t)?

Power (P) is defined as the rate of doing work or the rate of change of kinetic energy (K). For a block of mass m starting from rest on a smooth horizontal surface, the work-energy theorem states that the work done by the machine equals the change in kinetic energy:

P = dK/dt = d/dt (1⁄2 mv2)

Since the machine provides constant power, we can integrate both sides with respect to time (t):

∫ P dt = ∫ d(1⁄2 mv2)
Pt = 1⁄2 mv2

Solving for the speed (v):

v2 = (2P/m) t
v = √(2P/m) × √t

Since P and m are constants, the speed v is directly proportional to the square root of time (t), which is expressed as v ∝ √t. This matches the relationship described in option B.