A block is placed on a rough horizontal plane. A time-dependent horizontal force `F=kt^(2)` acts on the block. Here k is a positive constant. Acceleration -time graph of the block is

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the given force acting on the block and determine the acceleration as a function of time. The force is given as \( F = kt^2 \), where \( k \) is a positive constant.

### Step-by-Step Solution:

1. **Identify the relationship between force, mass, and acceleration**:
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = ma \]
Here, \( F \) is the force, \( m \) is the mass of the block, and \( a \) is its acceleration.

2. **Substitute the expression for force**:
We know that \( F = kt^2 \). Substituting this into the equation gives:
\[ kt^2 = ma \]

3. **Solve for acceleration**:
Rearranging the equation to solve for acceleration \( a \):
\[ a = \frac{kt^2}{m} \]
Here, \( \frac{k}{m} \) is a constant since both \( k \) and \( m \) are constants.

4. **Identify the form of the acceleration function**:
The expression \( a = \frac{k}{m} t^2 \) indicates that acceleration is proportional to the square of time \( t^2 \). This means that the graph of acceleration versus time will be a quadratic function.

5. **Determine the shape of the graph**:
Since the coefficient \( \frac{k}{m} \) is positive (as both \( k \) and \( m \) are positive), the graph of acceleration as a function of time will be a parabola opening upwards.

### Conclusion:
The acceleration-time graph of the block will be a parabola that opens upwards, indicating that as time increases, the acceleration increases quadratically.