To solve this problem, we need to understand how power delivered to a body in one-dimensional motion with constant acceleration depends on time.
Firstly, let's summarize the given situation:
The formula for power \( P \) in terms of force \( F \) and velocity \( v \) is:
P = F \cdot v
Since the motion is one-dimensional and the only force acting is due to acceleration, \( F = m \cdot a \), where \( m \) is the mass of the body.
The velocity \( v \) of the body at any time \( t \) when starting from rest is given by:
v = u + at
Where \( u \) is the initial velocity, which is 0. So,
v = at
Substituting these into the power formula, we have:
P = (m \cdot a) \cdot v = m \cdot a \cdot (at) = m \cdot a^2 \cdot t
This expression implies that power \( P \) is directly proportional to \( t \). Thus, the power delivered to the body increases linearly with time.
Therefore, the correct answer is:
Other options such as \( t^{\frac{1}{2}} \), \( t^{ \frac{3}{2}} \), and t^2 suggest different types of dependencies on \( t \), which are not applicable based on our derivation from the basic kinematics and power formulae.