The linear momentum of a particle varies with time as `p = a_0 +at+bt^2`. Which of the following graph represents force and time relation ?

Answer

To solve the problem, we need to analyze the given expression for linear momentum and derive the corresponding force as a function of time.

### Step-by-Step Solution:

1. **Understand the Given Momentum Equation**:
The momentum \( p \) of the particle is given by:
\[ p = a_0 + at + bt^2 \]
where \( a_0 \), \( a \), and \( b \) are constants.

2. **Relate Force to Momentum**:
According to Newton's second law, the force \( F \) acting on an object is the time derivative of its momentum:
\[ F = \frac{dp}{dt} \]

3. **Differentiate the Momentum Equation**:
We will differentiate \( p \) with respect to time \( t \):
\[ \frac{dp}{dt} = \frac{d}{dt}(a_0 + at + bt^2) \]
- The derivative of \( a_0 \) (a constant) is \( 0 \).
- The derivative of \( at \) with respect to \( t \) is \( a \).
- The derivative of \( bt^2 \) with respect to \( t \) is \( 2bt \).

So, we have:
\[ F = 0 + a + 2bt = a + 2bt \]

4. **Analyze the Force Equation**:
The expression for force \( F \) can be rewritten as:
\[ F = a + 2bt \]
This indicates that the force is a linear function of time \( t \) with a slope of \( 2b \) and a y-intercept of \( a \).

5. **Determine the Graph of Force vs. Time**:
Since \( F \) is a linear function of \( t \), the graph of force versus time will be a straight line. The y-intercept is \( a \) (the value of force when \( t = 0 \)), and the slope is \( 2b \).

### Conclusion:
The graph that represents the relationship between force and time will be a straight line starting from \( a \) on the y-axis and increasing (or decreasing) depending on the sign of \( 2b \).