A block of mass m is moving with a constant acceleration a on a rough horizontal plane. If the coefficient of friction between the block and ground is `mu`, the power delivered by the external agent in a time interval t from the beginning is equal to:

Options

A. `ma^(2)y`
B. `mu` mgat
C. `mu m ( a+ mu g)` gt
D. `m(a + mu g) `at

Explanation

To solve the problem step-by-step, we will analyze the forces acting on the block and calculate the power delivered by the external agent.

### Step 1: Identify the forces acting on the block
The block of mass \( m \) is moving on a rough horizontal plane with a constant acceleration \( a \). The forces acting on the block include:
- The external force \( F \) applied by the agent.
- The frictional force \( f \) opposing the motion, which can be calculated using the coefficient of friction \( \mu \) and the normal force (which is equal to \( mg \) for horizontal motion).

### Step 2: Write the equation of motion
Using Newton's second law, the net force acting on the block can be expressed as:
\[ F - f = ma \]
Where \( f = \mu mg \) (the frictional force). Therefore, we can rewrite the equation as:
\[ F - \mu mg = ma \]

### Step 3: Solve for the external force \( F \)
Rearranging the equation gives:
\[ F = ma + \mu mg \]
Factoring out \( m \) from the right-hand side:
\[ F = m(a + \mu g) \]

### Step 4: Determine the velocity of the block after time \( t \)
Using the kinematic equation for velocity:
\[ V = U + at \]
Assuming the initial velocity \( U = 0 \) (the block starts from rest), we have:
\[ V = at \]

### Step 5: Calculate the power delivered by the external agent
Power \( P \) delivered by the external force is given by the product of the force and the velocity:
\[ P = F \cdot V \]
Substituting the expressions for \( F \) and \( V \):
\[ P = (m(a + \mu g))(at) \]
This simplifies to:
\[ P = m(at)(a + \mu g) \]
Thus, the final expression for the power delivered by the external agent is:
\[ P = m(at)(a + \mu g) \]

### Final Answer
The power delivered by the external agent in a time interval \( t \) is:
\[ P = m(at)(a + \mu g) \]

---