Power=F.v. = m.a.v

The power delivered to a body is defined as the rate at which work is done on the body. In this case, the body is undergoing one-dimensional motion with constant acceleration.

The formula for power is given by P = Fv, where P is the power, F is the force acting on the body, and v is the velocity of the body.

Since the body is undergoing one-dimensional motion with constant acceleration, we can relate the force and velocity using Newton's second law of motion, which states that F = ma, where m is the mass of the body and a is the acceleration.

Therefore, we can rewrite the formula for power as P = mav.

Now, let's consider the relationship between the acceleration and time. The acceleration of the body is constant, so we can write a = Δv/Δt, where Δv is the change in velocity and Δt is the change in time.

Since the body is initially at rest, the change in velocity is just v, and the change in time is t. Therefore, we can rewrite the formula for acceleration as a = v/t.

Substituting this expression for acceleration into the formula for power, we get P = m(v^2)/t.

Now, we are given that the power delivered to the body is proportional to some function of time, t. Let's assume that the power is proportional to tn, where n is some constant.

Therefore, we can write P = ktn, where k is the proportionality constant.

Substituting this expression for power into the previous formula, we get ktn = m(v^2)/t.

Simplifying this equation, we get v^2 = ktn+1/m.

Taking the square root of both sides, we get v = (ktn+1/m)^0.5.

Since the velocity, v, is proportional to (tn+1/m)^0.5, we can conclude that the power delivered to the body is proportional to tc, where c = (n+1)/2.

Therefore, the correct answer is option 'B', tc.