The linear momentum `p` of a body moving in one dimension varies with time according to the equation `p=a+bt^(2)` where a and b are positive constants. The net force acting on the body is

Options

A. A constant
B. Proportional to `t^(2)`
C. Inversely proportional to `t`
D. Proportional to `t`

Answer

To solve the problem, we need to find the net force acting on the body given the linear momentum \( p \) as a function of time \( t \).

### Step-by-Step Solution:

1. **Understand the relationship between momentum and force**:
The force \( F \) acting on an object is related to the change in momentum \( p \) over time \( t \) by the equation:
\[ F = \frac{dp}{dt} \]
where \( p \) is the linear momentum.

2. **Substitute the given momentum equation**:
We are given that the momentum \( p \) varies with time according to the equation:
\[ p = a + bt^2 \]
where \( a \) and \( b \) are positive constants.

3. **Differentiate the momentum with respect to time**:
To find the force, we need to differentiate \( p \) with respect to \( t \):
\[ \frac{dp}{dt} = \frac{d}{dt}(a + bt^2) \]
Since \( a \) is a constant, its derivative is 0. The derivative of \( bt^2 \) with respect to \( t \) is:
\[ \frac{d}{dt}(bt^2) = 2bt \]
Therefore, we have:
\[ \frac{dp}{dt} = 2bt \]

4. **Determine the net force**:
Now, substituting back into the force equation:
\[ F = \frac{dp}{dt} = 2bt \]
This shows that the net force acting on the body is directly proportional to time \( t \).

### Final Answer:
The net force acting on the body is:
\[ F = 2bt \]